Preheat oven to 400°. Heat 1 tbsp. oil in a large ovenproof frying pan over high heat. Add onion, mushrooms, and garlic and cook, stirring, until softened and starting to brown, 4 minutes. Add lemon juice, spinach, and 1/4 tsp. salt; cook until slightly wilted, 4 minutes. Transfer to a small bowl.
Bring 4 cups water and 1/2 tsp. salt to a boil. Cover; turn off heat. Meanwhile, cut chicken breast from the side, stopping halfway through; unfold to lay flat. Place each between 2 pieces plastic wrap and, with a mallet or small saucepan, pound until 1/4 to 1/2 in. thick.
In frying pan, cook chicken in 2 batches, 1 tbsp. oil per batch, until brown on 1 side. Sprinkle each with 1/4 tsp. salt and transfer, uncooked side up, to a cutting board. Scoop 1/4 cup spinach mixture onto half of each breast, then top with a cheese slice. Fold chicken over filling like a taco shell; press to close.
Return chicken to pan (with juices) and bake until cooked through and cheese is melted, 10 minutes. Let sit 5 minutes.
Meanwhile, return water to a boil. Add polenta and cook, stirring constantly, until softened, 10 minutes. Stir in butter and rosemary. Divide polenta among 4 plates and top each with a chicken breast. Spoon broth over chicken.
Note: Nutritional analysis is per serving.