Set a large skillet over medium-high heat. Add the olive oil and sauté the spinach and mushrooms until wilted, about 5 minutes. Season with a bit of salt and black pepper. Stir in the tomatoes and cook until reduced by half, about 10 minutes.
Grease a 9x13-inch baking dish and set the chicken in a single layer. Season with salt and black pepper. Lay 2 slices of cheese on top of each chicken breast, and pour the sauce and veggies over the chicken.
Bake for about 25 to 30 minutes. Remove from oven and serve immediately.