Keep the heat until the skillet on medium-low. Add in the butter. Once melted, stir in the onion, carrots, celery and garlic with a pinch of salt and pepper. Stir in the thyme. Cook until softened, about 5 to 6 minutes. Use a wooden spoon to stir the vegetables and help scrape any brown bits of flavor from the bottom of the pot.
Stir in the mushrooms. Let them cook for about 5 minutes, until they soften. Stir in the tomato paste and cook for another 5 minutes.