Heat the oil in a large pot or Dutch oven over medium-high heat. Once the oil is hot, add the sliced onions, ½ teaspoon salt and ½ teaspoon pepper, stirring to combine.
Reduce the heat to medium and stir occasionally, until golden brown and jammy, 25 to 35 minutes. If you notice the onions starting to stick to the pan, add a splash of water or broth to deglaze. Stir in the garlic and thyme, and continue to cook until fragrant, 1 to 2 minutes.
Add the wine and balsamic vinegar, scraping up any brown bits at the bottom of the pot. Pour in the broth and pasta and season with the remaining teaspoon of salt and pepper, stirring to combine.
Increase the heat to high and bring to a boil. Once boiling, reduce the heat to low, cover, and simmer until the noodles are tender, stirring occasionally, 12 to 15 minutes.
Remove the lid. Stir in the mozzarella, heavy cream, and Dijon mustard. Continue to mix until the mozzarella cheese has melted throughout and the sauce is creamy. Taste and adjust seasonings, if necessary. Garnish with vegan parmesan and fresh minced parsley.
