Preheat the oven to 425 degrees F. Season the chicken generously with salt and pepper and sprinkle with 1 teaspoon thyme and the paprika. Melt 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the chicken and cook until browned, 2 to 3 minutes per side. Transfer to a baking sheet and roast until no longer pink in the middle, 8 to 10 minutes.
Meanwhile, melt the remaining 2 tablespoons butter in the skillet over medium-high heat. Add the onion, celery, carrot, remaining 1/4 teaspoon thyme and a pinch each of salt and pepper. Cook, stirring, until the vegetables are softened, about 5 minutes. Add the chicken broth and 3/4 cup water and bring to a boil. Stir in the gnocchi, breaking apart any that are sticking together. Cover and simmer until the gnocchi are just tender, about 5 minutes.
Uncover the skillet and stir in the peas and heavy cream. Cook until slightly thickened, about 1 minute; season with salt and pepper. Slice the chicken and serve with the gnocchi.