In a large deep skillet or Dutch oven, heat the butter and olive oil over medium heat until melted and hot.
Add the sliced onions, salt, pepper, and sugar. Cook, stirring frequently, for about 25 minutes until the onions are deeply caramelized and golden brown.
Add the minced garlic and fresh thyme leaves to the onions and cook for 1 minute until fragrant.
Sprinkle the flour over the onions and stir well to combine. Cook for 2 minutes to remove the raw flour taste.
Pour in the white wine and stir, scraping the bottom of the pan to deglaze and incorporate any browned bits. Let the wine reduce for 2 minutes.
Add the beef broth and whole milk to the pot and stir to combine.
Bring the mixture to a boil, then add the dried fettuccine pasta, pressing it gently into the liquid to submerge.
Reduce the heat to medium-low and simmer uncovered, stirring occasionally, for 12 to 15 minutes or until the pasta is cooked al dente and the sauce has thickened to a creamy consistency.
Remove the pot from heat and stir in the Gruyère and Parmesan cheeses until melted and smooth.
Taste and adjust seasoning with additional salt and pepper if needed.
Serve immediately, garnished with extra thyme or cheese if desired.