In a skillet, heat oil over medium-high heat. Cook mushrooms, onion, and thyme, stirring occasionally, until pale golden, about 5 min.
Spread butter over 1 side of each bread slice. In clean skillet, place 2 bread slices, butter-side down. Top each slice with one-quarter of the mushroom mixture and cheese. Top each portion with another bread slice, butter-side up.
Cook over medium heat until bottom bread is golden, 3 to 4 min. Using a flexible spatula, carefully flip and press to flatten slightly. Cook until cheese is melted and bottom bread is golden, about 1 min.
Repeat with remaining ingredients to make 2 more sandwiches.