Make the balsamic reduction: in a small saucepan combine 1/2 cup balsamic vinegar and 1 tablespoon honey. Simmer over medium-low, stirring occasionally, until thickened and reduced by a bit more than half, about 15 minutes. Remove from heat and set aside.
Prepare the chicken: place breasts on a cutting board, cover with plastic wrap, and lightly pound to an even 1/2-inch thickness. Alternatively, slice thick breasts lengthwise. Sprinkle both sides with 3/4 teaspoon of the kosher salt and 1/4 teaspoon of the black pepper.
Heat 1 tablespoon olive oil in a large heavy-bottomed skillet over medium-high heat. When hot, add the chicken, presentation-side down, and cook undisturbed 4 minutes until lightly browned. Flip and cook about 3 minutes more, or until the thickest part reaches 165°F and juices run clear; add short additional time if needed. Transfer chicken to a plate and cover to keep warm.
Without wiping the pan, lower heat to medium and add the remaining 1 tablespoon olive oil. Add the diced red onion and cook about 3 minutes, scraping up browned bits. Add the garlic and cook 30 seconds until fragrant.
Add the tomatoes, dried thyme, the remaining 1 tablespoon balsamic vinegar, 1 teaspoon honey, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Increase heat to medium-high and cook about 2 minutes until tomatoes begin to soften.
Return the chicken to the skillet, nestling it into the tomato mixture. Scatter the mozzarella over the top, cover the pan, and remove from the heat. Let sit 1–2 minutes to soften the cheese.
Uncover, drizzle the reserved balsamic reduction over the chicken and tomatoes, and sprinkle with the sliced basil. Serve warm.