1. Preheat oven to 400 degrees.

2. Place pumpkin in a roasting pan; rub cut sides with 3 tablespoons butter and place one sage leaf in the cavity of each pumpkin quarter; season with salt and pepper. Cover pan with parchment paper-lined aluminum foil. Transfer to oven and bake until pumpkin flesh is easily pierced with a fork, about 1 hour.

3. Remove from oven and discard foil. Let pumpkin stand until cool enough to handle. Discard sage leaves. Using a spoon, scoop out pumpkin flesh and transfer to a large bowl; discard skin. Using a fork, mash pumpkin until smooth; add mascarpone, Parmesan, and olive oil. Season with salt and pepper; stir to combine. Set filling aside.

4. Divide pasta dough into 4 equal pieces. Set the rollers of a hand-crank or electric pasta machine at their widest opening. Working with one piece at a time (and keeping the remaining pieces covered with plastic wrap), generously flour dough and run it through the pasta machine. Dust dough again with flour and fold into thirds like a letter. Run dough through pasta machine again. Repeat process 2 to 3 more times until smooth.

5. Now change the rollers of the pasta machine to the next decreasing setting, and roll out the dough once without folding. Keep rolling the sheet through the machine on decreasing settings until you have rolled it through the second to last setting. Transfer pasta to a lightly floured baking sheet and cover with plastic wrap. Repeat process with remaining pieces of dough.

6. Working with one pasta sheet at a time, cut into 3-by 3-inch squares. Place 1 tablespoon pumpkin filling toward the middle of the top half of each square. Using a wet pastry brush, brush edges of pasta to moisten. Fold bottom half of square over filling. Press air out and pinch dough together, working from the middle outwards to remove any air pockets. Seal edges with the ends of the tines of a fork; fold edges over filling and pinch together. Transfer cappellacci to a lightly floured baking sheet as you work.

7. Bring a large pot of water to a boil; generously salt water and return to a boil. Working in batches, add cappellacci; cook until tender, 5 to 6 minutes. Drain, reserving 1/4 cup pasta cooking liquid.

8. Meanwhile, tear remaining 6 sage leaves into small pieces. Heat 4 tablespoons butter and torn sage in a large skillet over medium high heat until aromatic and browned, taking care not to burn, about 3 minutes.

9. Reduce heat to low and add reserved pasta cooking liquid to skillet along with remaining 4 tablespoons butter; cook, stirring, until butter is emulsified. Add cappellacci to skillet and gently stir to combine. Serve in warmed bowls sprinkled with Pecorino Romano.