Instructions

1. Make the gingerbread cake: Position a rack in the center of the oven and preheat to 335°F. Grease three 6-inch square cake pans with nonstick spray and line the bottoms with parchment to fit. Spray again. Set aside.
2. In a large bowl, whisk together the flour, baking soda, cinnamon, ginger, salt, cloves, nutmeg, and black pepper. Set aside.
3. In a medium bowl, whisk together the molasses, vanilla, and eggs. 
4. In a stand mixer fitted with the paddle attachment, beat the dark brown sugar and oil on low until combined. Add the egg mixture and continue beating until incorporated, pausing and scraping the sides and bottom of the bowl with a silicone spatula as needed. Beat in one-third of the flour mixture, followed by the buttermilk. Beat in another third of the flour mixture, followed by the sour cream. Finally, beat in the remaining flour mixture. Once again, scrape the bottom and sides of the bowl, then continue beating on medium until the batter is smooth and homogeneous. 
5. Divide the batter evenly among the prepared cake pans, then bake, rotating the pans halfway through, until the tops are domed and a tester inserted into the center comes out clean, 30–35 minutes. Set aside on wire racks to cool completely. Clean and dry the stand mixer bowl and paddle.
6. Meanwhile, make the butterscotch: In a large heavy pot over medium heat, stir together the dark and light brown sugars and the butter and cook, stirring frequently, just until melted, about 9 minutes. (Do not caramelize.) Turn the heat to medium-high and add the vanilla bean, if desired. Add 1 cup of the cream and simmer, stirring frequently to prevent it from boiling over, until reduced slightly, 4–6 minutes. Continue adding the remaining cream in 1-cup increments, simmering and stirring frequently until slightly reduced, 4–6 minutes between additions. Remove from heat, then stir in the salt and vanilla extract. Remove the vanilla bean (if using) and reserve for another use, then measure 1 cup of the butterscotch for the frosting and set aside to cool to room temperature. Reserve the rest for another use or transfer to clean, heatproof jars, cover tightly, and store in the fridge for up to 1 month. 
7. Make the frosting: In the stand mixer fitted with the paddle attachment, beat the cream cheese and butter on medium until homogeneous, about 1 minute. Add the confectioners sugar and beat on low for 1 minute, then turn the speed to medium and continue beating until smooth, about 1 minute more. Using a silicone spatula, scrape down the sides and bottom of the bowl, then beat in the reserved 1 cup of butterscotch, the salt, and vanilla until smooth and fluffy, about 2 minutes more. Transfer to a medium bowl, cover, and set aside at room temperature. Clean and dry the stand mixer bowl and paddle.
8. Make the gingersnap dough: In the stand mixer fitted with the paddle attachment, beat the butter briefly on medium, 5–10 seconds. Add the dark brown sugar and beat until light and fluffy, about 4 minutes. Beat in the molasses, vanilla extract, and egg, scrape down the bottom and sides of the bowl, then beat on medium until nearly homogeneous, 20–30 seconds.
9. In a medium bowl, whisk together the flour, baking soda, ground and fresh ginger, cinnamon, both salts, the cloves, nutmeg, and black pepper, then add the flour mixture to the mixer bowl all at once. Beat on low just until a shaggy dough forms, about 30 seconds. Turn the dough out onto a large sheet of parchment, then gently press it into an even ½-inch rectangle. Top with a second sheet of parchment, then roll the dough into a 13- by 18-inch rectangle. Refrigerate the dough while you assemble the cake.
10. Unmold 2 of the cakes and leave the third in its pan. Slide a 6-inch strip of acetate along each side of the third pan. (Eventually, this will be the top layer of the cake.) Add ½ cup of the reserved frosting and, using a small offset spatula or the back of a spoon, spread evenly over the surface of the cake. Add a second layer of cake, bottom side up (be sure to remove this layer’s parchment if it has stuck to the cake), press down gently, then top with another ½ cup of frosting. Continue with the third cake round, adding another ½ cup of frosting, then refrigerate the cake until firm, at least 4 hours or up to 24. (If refrigerating for more than 4 hours, cover tightly in plastic wrap.)
11. Meanwhile, make the toffee: Line a baking sheet with a silicone baking mat. In a small heavy pot over medium heat, melt the butter. Stir in the sugar, corn syrup, salt, vanilla, and 1 tablespoon of water. Turn the heat to low and cook, stirring occasionally, until the syrup begins to brown, about 4 minutes. Continue cooking, stirring frequently, until the mixture turns a deep golden brown and just begins to smoke, 3–4 minutes. Turn off the heat, stir a couple of times to cool slightly, then pour the toffee onto the silicone baking mat. Set aside for a few seconds, just until cool enough to touch, then stretch the toffee to create long, imperfect shards. Set aside until completely cool and firm, about 15 minutes.
12. Position a rack in the center of the oven and preheat to 325°F. 
13. Retrieve the gingersnap dough from the fridge and, using a ruler, score the rectangle into 1-inch-wide strips. Measure the height of the assembled layer cake and score the gingersnap strips to match that height. (These will be the “boards” of the gingerbread house.) Arrange the strips on a parchment-lined baking sheet, spacing at least 2 inches apart. Bake until the gingersnaps are light golden, puffed, and barely set, about 8 minutes. Transfer the baking sheet to a wire rack to cool.
14. Invert the layer cake onto a serving plate; remove and discard the acetate strips. Using an offset spatula or the back of a spoon, spread the bottoms of the gingersnaps with a little frosting, then adhere the cookies to the sides of the cake to fully cover. Scrunch a piece of clean parchment and place on the top of the cake. Layer the remaining gingersnap cookies over the parchment to cover, creating a “roof” for the house. Spread the backs of the toffee pieces with a bit more frosting, then, starting from the bottom edge of the roof and working upward, adhere the toffee to the gingersnaps, creating a shingled effect. Sprinkle the roof with confectioners sugar to resemble snow. If desired, paint the gingerbread walls with some of the leftover frosting to create a snowy, white-washed effect, or pipe on a few architectural details. Serve at room temperature.