1. In a small saucepan, cook the squash purée over medium heat, stirring frequently, until it stops releasing moisture and is reduced to 1 1/4 cups, 10 to 12 minutes. Transfer to a small bowl to cool.

2. In a small saucepan, melt 2 tablespoons butter over medium heat. Add shallots; cook, stirring, until soft and beginning to brown, 3 to 5 minutes. Add chopped sage; cook, stirring, 1 minute more. Remove from heat; combine with squash. Season with salt and pepper; stir in nutmeg. Let cool completely.

3. Generously dust two baking sheets with semolina; set aside. Divide pasta dough into 4 pieces; cover unused pieces with plastic wrap to keep them from drying out. Dust a piece of dough lightly with flour. Using a pasta maker, roll dough through the widest opening. Fold dough into thirds, and pass through machine again, layered side first. Repeat process three or four more times, until dough is smooth. Continue running the dough through the remaining settings, using additional flour sparingly, until pasta sheet is very thin. Dough should be at least 5 inches wide.

4. Place dough on a lightly floured surface; halve crosswise. Cover half with plastic wrap. On the other half, place scant tablespoons of squash filling 1 inch apart in 2 rows. Using a pastry brush dipped in water, lightly moisten pasta around each mound of filling. Top with the remaining half sheet of pasta; press around mounds to eliminate air inside and to seal. Cut pasta into 2 1/2-inch squares. Brush away excess flour. Place ravioli on prepared baking sheets. Repeat with remaining dough and filling.

5. For fried sage leaves: In a small saucepan, heat oil over medium-high heat. Drop a few sage leaves at a time into the hot oil, and fry until oil around leaves stops bubbling but before leaves brown, about 5 seconds. Using a slotted spoon, transfer to a paper-towel lined plate; set aside.

6. Bring a large stockpot of water to a boil. Add salt and ravioli. Gently stir once, and cook at a gentle boil until the ravioli have floated to the top and are just tender, 3 to 5 minutes. Using a slotted spoon, divide evenly among four plates.

7. In a small saucepan, melt the remaining 4 tablespoons butter over medium heat, and cook until lightly browned and aroma is nutty, 6 to 8 minutes. Drizzle over ravioli, and garnish with cheese and fried sage leaves. Serve immediately.