Place chicken breasts between 2 sheets of heavy plastic on a solid, level surface; firmly pound with the smooth side of a meat mallet to a 1/4-inch thickness. Season with salt and pepper.

Heat 1 tablespoon olive oil in a skillet over medium heat. Add mushrooms and garlic; cook and stir until softened, about 5 minutes.

Preheat oven to 400 degrees F (200 degrees C).

Combine mushroom-garlic mixture, cream cheese, spinach, green onions, and red pepper flakes in a bowl.

Spoon 1 tablespoon cream cheese mixture into the middle of each chicken breast. Roll chicken breasts around mixture and secure with toothpicks.

Heat remaining 3 tablespoons in an ovenproof skillet over medium heat. Arrange chicken rolls in the skillet in a single layer; cook until golden, about 5 minutes per side.

Transfer skillet to the preheated oven; bake until chicken is no longer pink in the center, 10 to 15 minutes more.