Pat chicken cutlets dry with a paper towel, then drizzle with olive oil. Season on both sides with dried oregano, kosher salt, and black pepper.
Preheat a broiler to high. Heat a large skillet over medium-high heat, then add unsalted butter. When hot, add seasoned chicken cutlets and cook undisturbed until golden brown. Flip each cutlet over, then cook. Transfer the chicken to a plate.
Stir in white wine, balsamic vinegar, honey, and fresh basil leaves. Return the chicken to the pan, then place cover each piece with sliced mozzarella.
Transfer the skillet under the broiler, then cook until the cheese is melted and bubbly. Keep a close eye on it so the cheese doesn’t burn and rotate the pan halfway through.
Sprinkle 2 Tbsp capers on top, then garnish with a little grated Parmigiano Reggiano. Serve the chicken with some of the tomatoes and sauce spooned on top. Enjoy!