Add 3 T. of the butter to a Dutch oven over medium heat. When melted, add the sliced onions, sugar, and 1 tsp. of salt. Stir and cover. Cook covered for 20 minutes, stirring occasionally.
Remove the lid and continue cooking for 40 to 50 minutes, lowering the heat if the onions begin to scorch. When done, the onions should be a deep brown and have a jammy consistency.
While the onions cook, add 1 T. butter and 1 T. olive oil to a pan over medium-high heat. Add the sliced mushrooms, thyme, and pepper, and cook until softened, 5-7 minutes. Stir in the remaining ½ tsp. of salt and set aside.
Unfold the thawed puff pastry onto a sheet of parchment paper. Roll out the pastry with a rolling pin to remove creases and even out.
With a fork, poke holes all over the puff pastry leaving a small border for the crust. Brush the entire sheet with the remaining olive oil.
Spread the carmelized onions over the puff pastry, leaving a small border. Spread the mushrooms evenly across (you may not use all of them). Using your fingers, drop small pieces of goat cheese all over the tart.
Move the parchment paper with tart to a baking sheet. Bake for 20-25 minutes, or until the crust is browned.
Add the balsamic vinegar and brown sugar to a small saucepan over medium-high heat. Bring to a boil. Reduce heat to medium-low and cook until sauce is thickened, 10-15 minutes. You want it to coat the back of a spoon.
When done, remove the tart from the oven and allow to rest for 5-10 minutes. Drizzle with the balsamic glaze, slice into squares, and serve.