Fried Egg with Hazelnuts, Chanterelles, Green Garlic, and Blackberries

Heat oven to 350°. Toss hazelnuts with oil and a pinch of salt on a parchment paper-lined baking sheet; bake until dark and fragrant, 10–12 minutes. Cool, then lightly crush; set hazelnuts aside. Mix blackberries, 1 tsp. each lemon juice, salt, and sugar and set aside. Heat 2 tbsp. butter in a 12-inch cast-iron skillet over medium-high and cook until dark and nutty, 3 minutes. Add chanterelles, beans, green garlic, garlic cloves, thyme, salt, and pepper and cook until green garlic is caramelized, 7 minutes. Toss with remaining lemon juice and keep warm. Meanwhile, heat remaining butter in a 12-inch nonstick skillet over medium. Add eggs and 1 tsp. water and season with salt and pepper; cook, covered, until whites are set but yolks are still runny, about 3 minutes; divide between 4 plates. Arrange vegetables around eggs and finish with reserved blackberries and hazelnuts, and the herbs.