Bring a large pot of water to a boil over high heat. Add the pasta and cook according to package directions for al dente, stirring occasionally. Drain the pasta, reserving about 2 cups of the pasta cooking liquid.
While the pasta is cooking, warm 1 tablespoon of oil in a large pot over medium-high heat. Add 1 tablespoon of butter and allow it to melt. Add the onions to the pot and toss to coat. Cook the onions, stirring constantly, until they begin to soften, about 5 minutes.
Sprinkle with sugar, about ½ teaspoon of salt, and pepper. Continue cooking for 10 more minutes, until the onions are deeply caramelized. Add the garlic and cook about 30 seconds more until fragrant. Add the wine and scrape up any browned bits on the bottom of the pan. Add the fresh thyme and the cream.
Bring to a low simmer and add the cooked pasta to the sauce. Stir to coat and then sprinkle with Parmesan. If the pasta appears dry, add a splash or two of the reserved pasta water. Taste the pasta and add more salt, only if needed.
Serve warm, topped with additional Parmesan, pepper, and parsley.