Preheat the oven to 425°F.
Melt the 2 teaspoons of butter in a wide pan over medium heat. Stir in the onions.
Sprinkle in the brown sugar, 1/4 teaspoon of salt, baking soda, and vinegar. Cook for a couple of minutes while stirring.
Put a lid on the pan and reduce the heat to medium-low. Allow the onions to sweat for about 20 minutes, stirring every 5 minutes or so. The onions are ready when they're really soft and translucent.
Move the onions to a bowl and set aside. Do not rinse or wipe out the pan!
In the same pan, over medium heat, melt the 1 1/2 tablespoons of butter and stir in the flour. Allow to cook for about a minute while constantly stirring.
Stir in the white wine and let it simmer until the wine is almost all the way reduced. The roux (flour and butter) will look thick and smooth when it's ready.
Whisk in the stock and milk until it's all smooth.
Stir in the thyme and dry pasta. Bring everything to a boil, then reduce it to a simmer. Cover the pan with a lid and cook the pasta until it's al dente, stirring every few minutes. You may add some water if the sauce is looking too thick or dry. How long this takes/ how much liquid you need will depend on the kind of pasta you're using as well as your elevation (with the smaller pasta I used at sea level, it only took 7 minutes with no additional liquid).
Turn off the heat and stir in the goat cheese, mustard, 1/2 teaspoon of salt, and the onions until the cheese is all melted.
Transfer the pasta to a 9x13 pan and sprinkle with the Romano cheese.
Bake in the preheated oven for 10 minutes, or until the cheese on top starts to get toasty.