Heat a large nonstick skillet over medium-high heat. Coat pan with cooking spray. Add garlic and turkey to pan; cook 5 minutes or until turkey is no longer pink, stirring to crumble. Add crushed red and ground red peppers; cook 1 minute. Spoon turkey mixture into a bowl.
Recoat pan with cooking spray; return pan to medium-high heat. Add 1/2 teaspoon salt and mushrooms to pan; sauté 5 minutes. Stir in peas; cook 1 minute. Stir in turkey mixture; cook 3 minutes or until thoroughly heated. Cover and set aside.
Bring a large saucepan of water to a boil. Add gnocchi to pan; cook just until gnocchi begin to float. Drain gnocchi, reserving 1/2 cup cooking liquid. Combine turkey mixture, remaining 1/4 teaspoon salt, 1/2 cup cooking liquid, and gnocchi in a large bowl; toss well. Sprinkle with cheese and basil. Serve warm.