In a small bowl, whisk together balsamic vinegar, honey, olive oil, garlic, Italian seasoning, onion powder, garlic powder, and Dijon mustard. Set aside half of the mixture for glazing later.
Season the chicken breasts with sea salt and black pepper, then coat with the balsamic marinade. Let it sit for at least 30 minutes (or up to 2 hours for deeper flavor).
Heat vegetable oil in a large skillet over medium-high heat. Add the chicken and cook for 5-6 minutes per side, until golden brown and cooked through. Remove and set aside.
In the same skillet, add grape tomatoes and sauté for 2-3 minutes until softened.
Return the chicken to the skillet, top with mozzarella slices, and drizzle with the reserved balsamic glaze.
Cover the skillet for 1-2 minutes, allowing the cheese to melt slightly.
Remove from heat, garnish with fresh basil, and serve warm.