Preheat the oven to 400°F, positioning a rack in the center. In a 10-inch cast-iron or deep-sided skillet over medium heat, warm the olive oil. Season the chicken on both sides with ½ teaspoon of salt and ½ teaspoon of pepper. Sear the chicken until golden, approximately 5 minutes per side, then transfer to a baking sheet.
Bake the chicken in the preheated oven until an instant-read thermometer inserted into the thickest portion registers 165°F, about 15–18 minutes. Allow the chicken to rest briefly on a cutting board before slicing; keep warm. Maintain the oven temperature for later use.
In the same skillet, sauté the shallot over medium heat until softened, about 3 minutes, stirring occasionally. Incorporate the garlic and cook for an additional minute, stirring, until fragrant. Season with ¼ teaspoon of salt.
Deglaze the skillet with white wine, stirring for approximately 1 minute. Stir in the tomato sauce and chicken broth, allowing the mixture to thicken and bubble gently, around 3 minutes. Add the gnocchi and cook for 1–2 minutes, just until tender, seasoning with ½ teaspoon of salt. Introduce the halved cherry tomatoes and cook for an additional 2–3 minutes until they begin to soften. Adjust seasoning with the remaining salt if necessary.
Activate the broiler. Nestle the chicken slices in the center of the skillet and scatter the cubed mozzarella over the top.
Broil for approximately 2 minutes, monitoring closely, until the mozzarella has melted completely.
Garnish with fresh basil leaves and finish with a drizzle of balsamic glaze before serving.
