Bring a large pot of heavily salted water to a rolling boil. Add the spaghetti and cook according to package directions until al dente. Before draining, reserve 1.5 cups of the starchy pasta water. Drain the pasta and set aside.

While the pasta cooks, heat the olive oil and butter in a large skillet or Dutch oven over medium-low heat. Add the thinly sliced shallots and a pinch of salt. Cook, stirring occasionally, for 15-20 minutes, or until the shallots are deeply golden brown, very soft, and sweet. Do not rush this step; low and slow heat is key for proper caramelization.

Add the minced garlic and red pepper flakes (if using) to the skillet. Cook for another 1-2 minutes until fragrant, being careful not to burn the garlic.

If using, pour in the white wine to deglaze the pan, scraping up any browned bits from the bottom. Let it simmer for 1-2 minutes until slightly reduced. If not using wine, skip this step.

Stir in the vegetable or chicken broth and balsamic vinegar. Bring to a gentle simmer and let it cook for about 3-5 minutes, allowing the flavors to meld and the sauce to reduce slightly.

Add the cooked spaghetti directly to the skillet with the shallot sauce. Pour in about 1 cup of the reserved pasta water. Toss vigorously with tongs, allowing the pasta to absorb the sauce and for the starch from the pasta water to create a creamy emulsion.

Stir in the grated Parmesan cheese until melted and incorporated. If the sauce seems too dry, add a little more reserved pasta water, a tablespoon at a time, until it reaches your desired consistency. Taste and adjust seasoning with salt and freshly ground black pepper.

Garnish generously with fresh chopped parsley and serve immediately with extra Parmesan cheese on the side.