Combine 1/2 cup balsamic vinegar and 1 tablespoon honey in a small saucepan. Simmer over medium-low heat, stirring occasionally, until thickened and reduced by just over half, about 15 minutes. It should coat the back of a spoon. Remove from heat and set aside.

Place chicken breasts on a cutting board, cover with plastic wrap, and pound to an even 1/2-inch thickness. Season both sides with 3/4 teaspoon salt and 1/4 teaspoon pepper.

Heat 1 tablespoon olive oil in a large heavy-bottomed skillet over medium-high heat. Add the chicken top-side down and cook undisturbed for 4 minutes until golden brown. Flip and cook for another 3 minutes or until an instant-read thermometer reads 165 degrees F at the thickest part. Transfer to a plate and cover loosely.

Reduce heat to medium. Add remaining 1 tablespoon olive oil. Cook the red onion for about 3 minutes, scraping up the browned bits from the pan. Add garlic and stir for 30 seconds until fragrant.

Add tomatoes, thyme, remaining 1 tablespoon balsamic vinegar, 1 teaspoon honey, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Increase heat to medium-high and cook for about 2 minutes until tomatoes begin to soften. If excess liquid remains, cook one additional minute uncovered.

Nestle the chicken back into the skillet and scatter mozzarella throughout. Cover the pan, remove from heat, and let stand 1 to 2 minutes until the mozzarella is soft and melty.

Drizzle the balsamic reduction over the top and sprinkle with fresh basil. Serve warm straight from the skillet.