Combine flour, salt and pepper in a shallow dish.
Place chicken between two large pieces of plastic wrap. Pound with the smooth side of a meat mallet or a heavy saucepan until the chicken is an even ¼-inch thickness.
Dredge the chicken in the flour mixture.
Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat.
Add spinach and cook, stirring often, until wilted, 1 to 2 minutes. Transfer to a small bowl.
Add the remaining 2 teaspoons of oil to the pan. Add the chicken, and cook until golden on first side, 2 to 3 minutes.
Turn the chicken, reduce heat to medium, top with the wilted spinach, marinara sauce and Parmesan.
Sprinkle with mozzarella, cover and cook until the cheese is melted and the chicken is cooked through, 2 to 3 minutes.
Serve on rolls.