Finely grate Parmesan cheese and garlic into a medium bowl. Add pumpkin purée, egg, 1/2 teaspoon kosher salt, 1/4 teaspoon black pepper, and nutmeg. Whisk until well combined.

Crumble goat cheese into the bowl and gently fold until combined. Cover with plastic wrap and refrigerate until ready to use.

Line 2 baking sheets with parchment paper and dust with flour. Divide pasta dough into 4 portions and roll each through pasta rollers until very thin, then cut in half crosswise to have 8 sheets.

On a floured surface, place 2 sheets horizontally. Drop 1-teaspoon portions of filling about 1 inch from the top and 2 to 2 1/2 inches apart on the sheets.

Dampen fingers in water and trace around the edges of each sheet and between portions of filling. Fold the bottom half of each sheet over the filling, pressing out excess air and sealing together.

Trim edges with a fluted pastry wheel cutter or sharp knife to form individual ravioli. Transfer to the second baking sheet. Repeat with remaining sheets.

Bring a large pot of salted water to a boil. Cut butter into 3 pieces. Pick sage leaves and pat dry. Coarsely chop pecans.

Toast pecans in a frying pan over medium-high heat until dark golden brown, about 6 to 7 minutes. Transfer to a bowl.

Melt butter in the pan over low heat, then add sage leaves and cook until crisp and dark, about 3 minutes. Remove sage to a paper towel-lined plate and keep butter in the pan.

Cook ravioli in boiling water until floating, about 3 to 4 minutes. Reserve 1/2 cup pasta water, then transfer ravioli to the pan with the butter.

Return pan to medium heat. Add 1/4 cup pasta water, remaining 1/4 teaspoon kosher salt, and 1/4 teaspoon black pepper. Simmer while gently tossing until sauce thickens, about 2 minutes.

Transfer ravioli and sauce to a serving dish. Top with pecans, fried sage, and more grated Parmesan cheese.