Preheat the oven to 375°F and lightly grease a baking dish with a small amount of olive oil.

Heat 1 tablespoon of olive oil in a skillet over medium heat. Add the mushrooms and cook until they release their liquid and start to brown. Stir in the garlic and spinach, cooking just until the spinach wilts. Remove the pan from heat and mix in the mozzarella, Italian seasoning, salt, and pepper.

Place each chicken breast between sheets of plastic wrap and pound to an even 1/2-inch thickness. Spoon the mushroom and spinach mixture evenly across each piece, then roll the chicken tightly around the filling and set seam side down in the prepared baking dish.

Brush the tops of the roll-ups with the remaining tablespoon of olive oil and sprinkle lightly with any extra salt and pepper if desired. Bake for 25 to 30 minutes until the chicken reaches an internal temperature of 165°F and the cheese is bubbling inside.