Place the 2 1/4 cups flour in a mound on a work surface. Make a well in the center large enough to hold the beaten eggs and pour the eggs into the well. Using a fork, begin gradually incorporating some of the flour from the sides, taking care not to break the flour wall. When the eggs are no longer runny, you can stop worrying about the wall. Continue working in more flour until the dough is no longer wet.

Begin kneading the dough by hand, adding as much additional all-purpose flour as needed until the dough is smooth and no longer sticky, 3 to 5 minutes.

Dust baking sheets with flour. Divide the dough in half. Keep one half on the work surface, covered with a kitchen towel to prevent it from drying. Set up your pasta machine alongside another work surface. Lightly flour the work surface with some of the reserved sieved flour. Using a rolling pin, flatten the other dough half into a rectangle thin enough to go through the rollers at the widest setting. Pass the dough through the rollers once, then lay the resulting ribbon down on the work surface and flour it lightly. Fold into thirds lengthwise to make a rectangle and flour both sides lightly. Flatten the dough with the rolling pin until it is thin enough to go through the rollers again. With one of the two open edges going first, pass the dough through the rollers nine more times at the widest setting; after each time, flour, fold and flatten the dough as described. After 10 trips through the wide rollers, the dough should be completely smooth and supple.

Now you are ready to thin the dough. Starting at the second-to-widest setting, pass the dough through the rollers repeatedly, setting the rollers one notch narrower each time. When the pasta ribbon gets unwieldy, cut it in half and continue rolling one part at a time until the dough reaches the desired thinness.

Arrange the finished pasta sheets on the prepared baking sheets and cover with kitchen towels to prevent drying. Repeat the entire process with the second half of dough.

Preheat oven to 400 degrees. Cut squash in half, remove seeds, and drizzle with 1 tbsp olive oil. Place flesh side down on baking sheet, and bake until very soft, about 40 minutes.

In a dry pan over medium heat, lightly toast chopped hazelnuts. Set aside. In the same pan, add remaining tablespoon of olive oil, and cook onion with sage until soft and golden. Scoop flesh from cooked squash, and add to onion mixture and mash together. Allow to cool slightly, then add mascarpone and combine. Season with salt and pepper to taste.

Drop teaspoonfuls of squash mixture about an inch apart the length of a sheet of pasta. Fold pasta over filling, and use dampened fingers to seal the entire length. Press pasta down around all sides of filling to seal. Use a pizza cutter or sharp knife to cut the pasta into squares of equal size, each with a dollop of filling in the center. Fold the cut edges under, around all four sides of ravioli. Use a fork to crimp and seal. Repeat with remaining pasta sheet. Drop ravioli into a pot of gently boiling water and cook until they are tender, and rise to the surface (about 7 minutes).

Heat butter and hazelnuts in a pan over medium heat until butter browns. Add broth, and remove from heat. Drizzle brown butter sauce over ravioli on plate, and top with *shaved Parmigiano-Reggiano, if desired.

*A vegetable peeler works great to make shavings and curls.