Instructions

1. Combine both flours in a stand mixer fitted with a paddle. With the motor running, add egg yolks, one at a time, beating well after each addition. Add oil and 3 tbsp. water; mix until dough forms. Transfer dough to a lightly floured surface; knead until smooth, about 5 minutes. Divide dough into 2 disks and wrap in plastic wrap; let rest 30 minutes.
2. Heat oven to 400°. Toss squash, butter, salt, and pepper on a baking sheet; spread into an even layer. Bake until squash is tender and golden, about 30 minutes.
3. Bring a large pot of salted water to a boil. On a lightly floured surface and working with 1 disk of dough at a time, roll dough into an 8"-long oval; dust on both sides with flour. Using a pasta machine, pass dough through machine twice, using the widest setting. Using the next narrower setting, pass dough through machine twice more. Continue to roll dough, setting the rollers to the next narrower setting, until dough is &frac112;" thick. Lay dough sheet on a lightly floured surface and cut into six 4" squares. Add to boiling water; cook 20 seconds. Using a slotted spoon, transfer squares to a bowl of ice water until cold. Drain and spread dough onto paper towels to dry; transfer to greased baking sheets.
4. Heat oven to 500°. Divide caciocavallo between pasta squares; top with squash and the thyme. Bake until cheese is melted and golden brown, about 5 minutes. Heat oven to broil; broil 1 minute. Shave parmesan over the top.