To make ravioli filling: Preheat oven to 400°F and place a rack in the center. Arrange pumpkin, cut side down, on a parchment-lined rimmed baking sheet. Place in oven and roast until the pumpkin is tender when pierced with a paring knife, about 45 to 50 minutes. Remove from oven and allow it to cool. Using a spoon, scrape the roasted pumpkin flesh into a potato ricer and rice the pumpkin into a large bowl.
Place butter in a large skillet and heat over medium heat until butter foams. When foaming subsides, add shallots and cook, stirring frequently until soft, 4 to 5 minutes. Remove from heat and add shallot mixture and grated Parmesan cheese to pumpkin, stir well to combine. Taste and adjust seasoning with salt and pepper.
To make the ravioli: Lay one rolled out pasta sheet on a lightly floured work surface. Measure 2 inches in from one end of the pasta sheet and 2 inches down from the top edge and place 2 teaspoons of filling on that spot. Continue to drop 2 teaspoons of the filling, spaced 2 inches apart, until you come to the end of the pasta sheet. Using a small pastry brush, lightly brush pasta dough around and between the filling mounds with water. Lay a second sheet of pasta on top and press down around the filling to press out any air. Using a fluted pasta cutter or knife, trim the border of the dough and cut between the mounds to create ravioli. Repeat with the remaining pasta sheets and filling.
To make the brown butter sauce: Melt butter in a small skillet over medium-high heat, whisking occasionally, until the milk solids turn golden brown and the butter takes on a nutty aroma. Whisk in sherry vinegar, sage, and taste and adjust seasoning with salt and pepper.
To cook ravioli: Heat a large pot of water over high heat until boiling. Generously season the water with salt. Add the ravioli and cook until the pasta is al dente and the filling is heated through, about 4 minutes. Using a slotted spoon or spider, remove the ravioli from the water and transfer to serving platter. Spoon the brown-butter sauce over the top, season with salt and pepper and serve.