Bring a large pot of salted water to a boil and cook pasta according to package directions. Reserve 1/2 cup of pasta water before draining.
In a large skillet over medium-high heat, heat 1 tablespoon of olive oil. Add the sausage and cook, breaking it into crumbles, until browned and cooked through, approximately 5-7 minutes. Remove sausage with a slotted spoon and set aside.
In the same skillet, add the remaining oil and Brussels sprouts. Sauté cut-side down without stirring for 4 minutes to achieve caramelization, then toss and cook until tender, about 4 more minutes.
Add minced garlic and red pepper flakes to the skillet; cook for 1 minute until fragrant.
Return the cooked sausage to the skillet. Pour in the chicken broth to deglaze the pan, scraping up any browned bits from the bottom.
Toss in the cooked pasta and half of the Parmesan cheese.
Adjust consistency with reserved pasta water as needed. Season with salt and pepper.
Serve immediately, topped with the remaining Parmesan cheese.