Preheat oven to 400°F (200°C) and line a large baking sheet with parchment paper.

Toss the cubed butternut squash and halved Brussels sprouts with 2 tablespoons of olive oil, salt, and black pepper on the prepared baking sheet.

Spread vegetables in a single layer and roast for 25 to 30 minutes, tossing halfway through, until the squash is tender and the sprouts are charred at the edges.

While vegetables roast, bring a large pot of salted water to a boil. Cook the pasta according to package directions until al dente. Reserve 1/2 cup of pasta water before draining.

In a large high-sided skillet or cast-iron pan, heat the remaining 1 tablespoon of olive oil over medium-high heat. Add the sausage coins and sear for 3 to 4 minutes per side until deeply browned. Remove sausage from the pan and set aside.

Reduce skillet heat to medium. Add the butter to the rendered sausage fat. Once melted and bubbling, add the minced garlic, sage, and thyme. Sauté for 1 to 2 minutes until the garlic is fragrant and the butter begins to slightly brown.

Add the drained pasta and seared sausage back into the skillet. Sprinkle with smoked paprika and red pepper flakes.

Incorporate the roasted butternut squash and Brussels sprouts into the skillet. Gently toss everything together.

Add the grated Parmigiano-Reggiano and 1/4 cup of the reserved pasta water. Stir continuously for 1 minute until a light, glossy sauce coats the ingredients. Add more pasta water if the mixture appears too dry.

Season with additional salt and pepper to taste and serve immediately with extra cheese.