Heat olive oil in a large skillet over medium heat. Add the onions and sauté until softened and browned, about 5 minutes. Add the garlic and sauté for 1 minute more. Add the brussels sprouts, carrots, mushrooms and thyme. Increase heat to medium-high and let the vegetables sauté down until well softened, browned, and caramelized, about 15 to 20 minutes, stirring occasionally. Add salt and pepper as the vegetables cook down.
The mushrooms will release a lot of their moisture and then the mixture will brown and cook down.