Caramelize the onions: Melt butter and olive oil in a large pot or Dutch oven over medium heat. Add the sliced onions, salt, and pepper. Cook, stirring occasionally, for 20-30 minutes, or until the onions are deeply caramelized and golden brown. This is the most important step! Be patient and let the onions cook slowly. If they start to brown too quickly, reduce the heat.

Deglaze (optional): If using white wine, add it to the pot and scrape up any browned bits from the bottom. Let it simmer for a minute or two until slightly reduced.

Simmer the broth: Pour in the beef broth, heavy cream, thyme, and sugar. Bring to a simmer, then reduce heat to low and let it simmer gently for 15 minutes, allowing the flavors to meld.

Cook the pasta: While the sauce simmers, cook the pasta according to package directions until al dente. Reserve about 1/2 cup of the pasta cooking water before draining.

Combine and serve: Add the cooked pasta to the onion sauce. Stir in the Parmesan cheese and a little bit of the reserved pasta water if needed to thin the sauce to your liking. Stir well to coat the pasta evenly.

Garnish and enjoy: Serve immediately, garnished with fresh parsley and extra Parmesan cheese, if desired.