In a large skillet, heat the olive oil and butter over medium heat until melted. Add the sliced onions and sprinkle with sugar. Cook, stirring occasionally, for about 25-30 minutes, until the onions are golden brown and caramelized. If the onions start to stick, add a splash of water or broth to deglaze the pan.
While the onions are caramelizing, bring a large pot of salted water to a boil. Cook the pasta according to the package instructions until al dente. Reserve 1 cup of pasta water, then drain the pasta and set aside.
Add the minced garlic to the skillet with the caramelized onions and cook for 1-2 minutes until fragrant. Stir in the heavy cream and vegetable broth. Bring to a simmer and cook for 5-7 minutes, allowing the sauce to thicken slightly.
Stir in the grated Parmesan cheese and dried thyme. Season with salt and pepper to taste.
Add the cooked pasta to the skillet, tossing to coat the noodles evenly with the sauce. If the sauce is too thick, add some of the reserved pasta water until you reach your desired consistency.
Garnish with chopped fresh parsley and additional Parmesan cheese if desired.
