First prepare the crunchy topping. In a large skillet, melt the butter on mediumhigh heat. Cook until the milk fats start to separate and butter begins to lightly bubble and smell nutty. Whisk the butter until it develops a light brown color. Add the minced sage and fry for 30 to 40 seconds or until it is aromatic.
Add the chestnuts or walnuts and breadcrumbs. Stir until combined. Season with a generous pinch of salt. Set crunchy topping aside to cool.
Next, prepare the pasta and sauce. In a large pot, cook the cavatelli in salted water, according to the package instructions.
While the pasta cooks, wipe out the large skillet and coat with the olive oil. Fry the minced garlic for 30 seconds then add the shredded cabbage. Sauté until the cabbage softens, 2 to 3 minutes. Set aside.
Add the Parmesan, pumpkin purée and mascarpone cheese, and mix.
Strain the pasta when al dente, reserving ½ cup of pasta water. Transfer pasta to the skillet with the cabbage mix. Stir pasta until it all is coated with the pumpkin mascarpone cream. Add a few tablespoons of pasta water at a time to loosen up the sauce. Season with salt and pepper, keeping in mind the previously added pasta water has salt in it as well.
Serve pasta in a bowl. Add desired amount of crunchy topping and garnish with grated Parmesan.