To prep the mushrooms, turn the caps over and, using a spoon, gently scrape off the dark gills on the underside. Dice in a 1/2-in dice.
In a large, nonstick pan, heat the olive oil on high heat. Add the diced mushrooms and saute, tossing occasionally, until browned. Add the spinach and thyme and cook for a minute, stirring regularly, until spinach is nice and wilty. Remove from the heat. Season with kosher salt and pepper.
Heat another large, nonstick pan on medium-high heat. On a large cutting board, lay out 3 tortillas. Top each with a slice of cheese, then the mushroom-spinach mixture, then another slice of cheese. Top each with a tortilla.
Spray the heated pan with cooking spray. Carefully transfer the assembled quesadilla to the pan. Spray the top tortillas with cooking spray. Cook the quesadillas until nice and toasty on the underside, then carefully flip to toast the other side. Yield: 3 servings.