Prepare the Pasta:

1. Cook the Pasta: Bring a large pot of salted water to a boil and cook the pasta according to package instructions. Once cooked, drain and set aside, reserving about 1 cup of pasta water for later.

Caramelize the Onions:

2. Sauté the Onions: In a large skillet, heat 1 tablespoon of olive oil over medium heat. Add the thinly sliced onions and cook, stirring occasionally, for about 15-20 minutes until the onions become soft, golden brown, and caramelized. Season with a pinch of salt to help draw out the moisture from the onions.
3. Deglaze the Pan: Once the onions are caramelized, add the balsamic vinegar and stir, scraping up any browned bits from the bottom of the skillet. This adds extra flavor to the sauce.

Brown the Butter:

4. Brown the Butter: In a separate pan, melt the butter over medium heat. Continue cooking, swirling the pan occasionally, until the butter turns golden brown and gives off a nutty aroma, about 3-5 minutes.
5. Combine the Flavors: Once the butter is browned, add the thyme leaves and continue to cook for another 30 seconds to infuse the butter with flavor.

Make the Sauce:

6. Add the Broth: Pour the beef broth into the skillet with the caramelized onions. Stir to combine and bring to a simmer. Let the sauce cook for about 5 minutes to reduce slightly and intensify the flavor.
7. Combine with the Pasta: Add the cooked pasta to the skillet and toss to coat with the sauce. If needed, add a bit of the reserved pasta water to achieve the desired sauce consistency.

Finish the Dish:

8. Add the Cheese: Stir in the grated Parmesan cheese and shredded Gruyère until melted and the sauce is creamy. Taste and adjust the seasoning with salt and pepper.
9. Serve: Transfer the pasta to serving plates, garnish with fresh parsley, and sprinkle with more grated cheese if desired. For an extra kick, top with crushed red pepper flakes.