Bring a large pot of generously salted water to a boil. Cook the pasta until just shy of al dente. Before draining, reserve 1 cup of the pasta water. Drain the pasta and set aside.
While the pasta cooks, heat a large skillet over medium-high heat. Add the Italian sausage and cook, breaking it into bite-sized pieces, until browned and cooked through. Transfer the sausage to a plate, leaving a little of the rendered fat in the skillet.
Reduce the heat to medium. Add 1 tbsp butter and the diced onion to the skillet. Cook for 3-4 minutes, stirring occasionally, until softened. Add the garlic and cook for another 30-60 seconds, until fragrant.
Add the remaining 2 tbsp butter. Cook for 2-3 minutes, stirring or swirling the skillet frequently, until the butter begins to foam and the milk solids turn golden brown and smell nutty. Keep a close eye on it so the butter doesn’t burn.
As soon as the butter is browned, stir in the pumpkin puree to stop the butter from cooking further. Add the balsamic vinegar, salt, black pepper, and ground sage. Stir to combine and cook for 2–3 minutes.
Reduce the heat to low and stir in the heavy cream. Add the Parmesan and 1/2 cup of the reserved pasta water. Stir until the Parmesan has melted and the sauce is smooth.
Return the cooked sausage to the skillet, then add the pasta. Toss everything together until the pasta is evenly coated in the sauce. Add more reserved pasta water, a splash at a time, as needed until the sauce is silky and coats the pasta.
Taste and adjust the salt and pepper as needed. Serve topped with freshly cracked black pepper, red pepper flakes, and additional Parmesan cheese, if desired.