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Arrange a rack in center of oven; preheat to 400°. In a 10" cast-iron or high-sided skillet over medium heat, heat oil. Season chicken on both sides with ½ tsp. salt and ½ tsp. pepper. Cook, turning halfway through, until golden, about 5 minutes per side. Transfer to a baking sheet.
Transfer to oven and bake chicken until an instant-read thermometer inserted into thickest part registers 165°, 15 to 18 minutes. Transfer to a cutting board and let cool slightly. Slice and keep warm. Keep oven on.
Meanwhile, in same skillet over medium heat, cook shallot, stirring occasionally, until just softened, about 3 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more; season with ¼ tsp. salt.
Add wine and cook, stirring, to deglaze pan, about 1 minute. Add tomato sauce and broth and cook, stirring occasionally, until beginning to thicken and bubble, about 3 minutes. Add gnocchi and cook, stirring, just until tender, 1 to 2 minutes; season with ½ tsp. salt. Add cherry tomatoes and cook, stirring, until beginning to soften, 2 to 3 minutes; season with remaining ½ tsp. salt, if needed.
Turn on broiler. Arrange chicken in center of skillet and sprinkle with mozzarella.
Broil chicken, watching closely, just until mozzarella melts, about 2 minutes.
Top with basil. Drizzle with balsamic glaze.