Preheat oven to 400 degrees and line a baking sheet with foil.
Place butternut squash cubes in a resealable plastic bag and coat with 2 tablespoons of the olive oil, 1/2 teaspoon of the salt and 1/4 teaspoon of the pepper. Toss well and scatter in a single layer on prepared baking sheet.
Roast squash until tender and golden brown, about 25-30 minutes, tossing once halfway through.
While squash is roasting, prepare orecchiette pasta according to package instructions reserving 1 cup of the starchy cooking liquid; set pasta and cooking liquid aside.
Heat a large, deep pan over medium-high heat and coat with 1 tablespoon of the olive oil.
Remove sausage from casings and add to pan. Use a large wooden spoon to break up the sausage and begin browning the meat.
Stir in grated garlic when sausage is about halfway cooked through. Continue to cook until sausage is cooked through, stirring frequently, about 7-9 minutes.
Add reserved starchy cooking liquid and use a wooden spoon to deglaze brown bits from bottom of the pan.
Stir in torn kale and remaining 2 tablespoons of olive oil, 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook until kale is bright green, about 2-3 minutes.
Toss in roasted butternut squash, cooked pasta, Parmigiano-Reggiano and minced parsley. Check for seasoning and adjust accordingly.
Garnish with additional fresh minced parsley and grated Parmigiano-Reggiano if you wish.