Position a rack in the center of the oven; heat the oven to 200°F.
In a 10- or 12-inch nonstick skillet, heat the oil over medium high until hot. Add the mushrooms and cook, stirring occasionally, until they release most of their juices and are slightly browned, 5 to 7 min. Add the garlic, thyme, 1/2 tsp. kosher salt, and 1/4 tsp. pepper. Taste and adjust the seasoning as needed. Transfer the mushrooms to a medium bowl.
Spread the butter on one side of each tortilla and set the tortillas on a work surface, buttered side down. Distribute the Fontina among the tortillas, covering only half of each and leaving a 1-inch margin at the edge. Distribute the mushrooms on top of the cheese. Fold the tortillas in half to enclose the filling, creating a half-moon.
Wipe out the skillet with a paper towel. Over medium heat, cook two of the quesadillas, covered, until golden brown, about 4 min. Uncover and flip the quesadillas. Cook until the second side is golden brown and the cheese has melted completely, about 2 min. Transfer the quesadillas to the oven to keep warm and repeat with the remaining two quesadillas. (You can hold the cooked quesadillas in the warm oven for up to 30 min.) Cut the quesadillas in half (or smaller wedges if serving as an hors d’oeuvre) and serve.