Cook pasta according to package directions, omitting salt and fat. Drain in a colander over a bowl, reserving 1/4 cup pasta cooking water.
Heat a large skillet over medium-high heat. Coat pan with cooking spray. Sprinkle chicken with salt and pepper. Add chicken to pan; cook 3 minutes on each side or just until done. Remove chicken from pan; thinly slice across grain.
Return pan to medium heat. Add oil; swirl to coat. Add spinach, red pepper, and garlic; cook 3 minutes or until spinach wilts. Add cooked pasta, reserved pasta cooking water, chicken, lemon juice, and cheeses to pan; toss to coat. Cook 1 minute or until cheeses melt. Serve immediately.