Instructions

for the pasta

Process flour, eggs and yolks, and oil in food processor until mixture forms cohesive dough that feels soft and is barely tacky to touch, about 30 seconds. (If dough sticks to your fingers, add up to ¼ cup flour, 1 tablespoon at a time, until barely tacky. If dough isn't cohesive, add up to 1 tablespoon water, 1 teaspoon at a time, until it just comes together; process 30 seconds longer.)
Turn dough ball onto dry counter and knead until smooth, 1 to 2 minutes. Shape dough into 6-inch-long cylinder. Wrap in plastic wrap and let rest at room temperature for at least 1 hour or up to 4 hours. (Dough may be refrigerated for up to 24 hours.)

for the besciamella

Melt butter in 10-inch broiler-safe skillet over medium heat. Add flour, salt, pepper, and nutmeg and cook, stirring constantly, until golden and fragrant, about 1 minute. Slowly whisk in 2 cups milk and bring to boil. Reduce heat to simmer. Cook, whisking occasionally, until sauce is thick and smooth, about 5 minutes.
Transfer 1 cup besciamella to liquid measuring cup. Add water and remaining 1 cup milk to remaining besciamella in skillet and whisk until smooth. Set aside.
Cut pasta dough crosswise into 4 equal pieces. Working with 1 piece of dough (rewrap remaining dough), dust with flour, place on clean counter, and press into 3-inch square. Using heavy rolling pin, roll into 6-inch square. Dust both sides of dough lightly with flour. Starting at center of square, roll dough away from you in 1 motion. Return rolling pin to center of dough and roll toward you in 1 motion. Repeat steps of rolling until dough sticks to counter and measures roughly 12 inches long. Lightly dust both sides of dough with flour and continue rolling dough until it measures roughly 20 inches long and 6 inches wide, frequently lifting dough to release it from counter. (You should be able to easily see outline of your fingers through dough.) If dough firmly sticks to counter and wrinkles when rolled out, dust dough lightly with flour. Transfer pasta sheet to dry kitchen towel. Repeat with remaining dough.

for the nidi

Adjust oven rack to upper-middle position and heat oven to 375 degrees. Lay 2 pasta sheets on counter and trim short rounded sides of each sheet. Cut each sheet in half crosswise to create 4 rectangles. Arrange rectangles with short sides parallel to counter edge. Spread 2 tablespoons reserved thick besciamella over each sheet. Sprinkle evenly with half of fontina, followed by half of mushrooms. Lay half of ham over mushrooms, tearing slices to distribute evenly (ham will not completely cover surface). Starting at side nearest you, roll each pasta sheet into cylinder. Repeat with remaining pasta sheets, thick besciamella, fontina, mushrooms, and ham (do not wash measuring cup).
Transfer 1 cup thin besciamella to now-empty measuring cup and set aside. Cut each roll crosswise into 3 equal pieces and transfer, cut side down, to skillet, nestling pieces together. Drizzle reserved besciamella over rolls. Spray sheet of aluminum foil with vegetable oil spray and cover pan tightly. Transfer to rimmed baking sheet and bake until pasta is cooked through and sauce is bubbling, about 1 hour.
Remove sheet from oven and heat broiler. Remove foil and sprinkle evenly with Parmesan. Broil until tops of spirals are browned and crisp, 2 to 4 minutes. Let cool 10 minutes before serving.