Step 1

Preheat oven to 425°. Toss squash, shallot, garlic, oil, thyme, red pepper flakes (if using), and 4 sage leaves on a rimmed baking sheet until coated. Season with salt and pepper; cover loosely with foil. Bake until squash and shallot are very soft, 35–45 minutes. Transfer to a plate and let cool.

Step 2

Reduce oven temperature to 350°. Toast walnuts on baking sheet, tossing once, until golden brown and fragrant, 8–10 minutes. Let cool and finely chop; set aside.

Step 3

Peel shallot and garlic and discard herbs. Purée squash, shallot, garlic, Parmesan, marscapone, and lemon juice in a food processor until smooth. Scrape into pastry bag fitted with ¼" tip (or use a large resealable plastic bag and cut a small opening in 1 corner).

Step 4

Set pasta maker to thickest setting; dust lightly with cornmeal. Divide dough into 8 pieces. Working with 1 piece at a time and keeping remaining dough wrapped in plastic as you work, flatten dough into a narrow rectangle (no wider than mouth of machine); pass through rollers. Fold dough as needed to fit and roll again. Repeat without folding, adjusting machine to thinner settings after every pass and dusting with cornmeal if sticky, until pasta is 1/16" thick (setting 8 on most machines). (Alternatively, you can roll out sheets lengthwise with a rolling pin until 1/16" thick.)

Step 5

Lightly dust work surface with cornmeal. Working with 1 length at a time and keeping remaining dough wrapped in plastic as you work, arrange so long side is facing you. Starting 1" from short edge and 2" from long edge closest to you, pipe teaspoon-sized mounds of squash mixture down the length, spacing ¾" apart. Lightly brush water around each mound. Fold the long side closest to you over filling, extending at least 1" past filling, and press down length to seal. Using index fingers and thumbs on both hands, pinch dough on either side of filling, sealing dough and creating “pillow” shapes.

Step 6

Using pasta cutter or pizza cutter, trim long side of dough farthest from you about ½" from mounds, then trim short ends to create tidy edges all around. Discard trimmings. Cut between each mound of filling, making individual pasta. Transfer angolotti to a cornmeal-dusted sheet tray. Repeat with remaining dough and filling.

Step 7

Heat butter in a medium skillet over medium, swirling pan often, until butter foams, then browns, 4–6 minutes. Remove from heat and stir in remaining 16 sage leaves.

Step 8

Meanwhile, cook angolotti in a large pot of boiling salted water, stirring occasionally, until tender but slightly undercooked, 2 minutes. Drain, then add to skillet with brown butter along with ½ cup pasta cooking liquid. Cook over medium-high heat, tossing to coat, until most of the liquid has evaporated and sauce has thickened, 2–3 minutes. Season with salt if needed. Serve topped with grated Parmesan and reserved walnuts.

Step 9

Do Ahead: Angolotti can be made 3 months ahead. Freeze on sheet tray, then transfer to a resealable plastic bag and keep frozen.