With the pan on medium heat, add the butter to the pan. Once the butter melts, add the mushrooms and stir until softened and browned, about 5 minutes. Add the garlic and cook for 1 more minute. Stir in the broth and thyme, then bring to a simmer. Cook until the liquid is reduced by about half, 5 minutes. Stir in the heavy cream and mustard until combined, simmer until the sauce is thick enough to coat the back of a spoon, about 4 more minutes.