Season the chicken. Season chicken breasts on both sides with ¾ tsp Kosher salt and ¼ tsp black pepper. Preheat an oven to 375F.
Make the compound butter. Place 4 Tbsp softened, room temperature butter into a small mixing bowl. Add ½ tsp Kosher salt, 2 chopped garlic cloves, and 1 Tbsp chopped rosemary, then stir until well mixed.
Brown the chicken. Heat a large skillet over medium-high heat. When hot, add 2 Tbsp olive oil and 3 Tbsp of the garlic herb butter. Once melted, add the chicken cutlets and cook for 5 minutes on the first side, or until golden brown and seared. Flip and cook an additional 2-4 minutes on the second side. Remove the chicken and place on a plate, then add the chopped shallot and cherry tomatoes to the pan. Season with ½ tsp Kosher salt, then stir well. Allow to cook for 2 minutes, then pour in ¾ cup of chicken stock or white wine. Use a spatula to help release any brown bits from the bottom of the pan.
Finish in the oven. Return the chicken to the skillet, along with any juices, then place the skillet in a preheated oven and cook for 7 minutes, or until the internal temperature of the chicken is 165F.
Finish, then serve. Remove the chicken breasts once more, then stir in the baby spinach, mascarpone cheese, remaining 1 Tbsp of compound butter, and lemon zest. Taste and adjust seasoning of the sauce, if needed. Return the chicken to the pan, tossing to coat in the sauce, then serve with sprinkled chopped parsley and lemon wedges.