Heat the oil and butter in a large sauté pan over medium heat. Once the butter is starting to sizzle, add the mushrooms in a single layer. Let them cook for a few minutes undisturbed. Check the bottom of the mushrooms, and, when they're golden brown and caramelized on the edges, flip. Continue to cook until the other side is caramelized, as well. Add the onions and sauté until softened, about 2 minutes. Stir in the garlic.

Deglaze the pan with the wine and allow the liquid to cook off for another minute. Season with thyme, salt, and pepper to taste. Remove from heat.

Butter one side of each of the four slices of bread. Assemble the sandwiches with one piece of bread (buttered side out) followed by a slice of cheese, half the mushrooms, another slice of cheese, then a second slice of bread, buttered side out.

Heat a grill pan over medium heat. Add the sandwiches and grill for 3-5 minutes or until golden and crisp on one side. Carefully flip the sandwiches, cover the pan with a lid or cookie sheet, and cook an additional 2-3 minutes or until the opposite side is golden and cheese is extra melty. Serve immediately.