Season the breasts evenly on all sides with Italian seasoning, garlic powder, salt, and black pepper.
Heat olive oil in a large skillet over medium-high heat. Sear the meat for 5 to 7 minutes per side until golden brown and cooked to an internal temperature of 165°F. Remove from the skillet and set aside on a plate.
Lower the heat to medium. Add the vinegar and honey to the skillet, whisking to scrape up any browned bits from the bottom of the pan. Simmer for 2 to 3 minutes until the liquid reduces and thickens slightly.
Stir the halved cherry tomatoes into the glaze and cook for 1 minute just until they begin to soften.
Return the seared meat to the skillet. Top each piece evenly with mozzarella cheese. Cover the skillet with a well-fitting lid for 2 minutes, or until the cheese is completely melted.
Remove from heat, garnish generously with fresh basil, and serve immediately.