Preheat the oven to 350°F.
Rub the chicken breasts with 1 tablespoon of olive oil and season both sides generously with dried basil, oregano, salt, and pepper.
Heat a large, oven-safe skillet over medium-high heat; add the remaining oil, then place the chicken into the pan, searing swiftly on each side until golden brown. Transfer the browned chicken to a plate.
In the same skillet, combine the red onion, minced garlic, balsamic vinegar, and brown sugar, stirring and simmering the mixture for about 3 minutes to thicken slightly.
Return the chicken to the skillet, spooning some of the sauce over each piece. Transfer the skillet to the preheated oven and bake until the internal temperature reaches 165°F, approximately 10–20 minutes based on thickness.
Adjust the oven to broil setting.
Top each chicken breast with a slice of mozzarella and broil just until the cheese has melted, monitoring closely to prevent burning.
Immediately garnish with slices of fresh tomato and a sprinkle of chiffonade basil. Drizzle the reserved pan sauce over the top before serving.