In a large skillet over medium-low heat, melt butter with olive oil. Add shallot; cook, stirring, until soft, about 5 minutes.
Raise heat to medium. Add mushrooms; season with salt and pepper. Cook, covered, until mushrooms release their liquid, 5 to 6 minutes.
Uncover; raise heat to high. Cook, tossing, until liquid evaporates and mushrooms are brown, 3 to 5 minutes.
Add thyme and wine. Cook until skillet is almost dry, 1 minute. Serve hot.