Pat chicken breasts dry with paper towels. If uneven in thickness, place between plastic wrap and gently pound to an even 3/4-inch thickness. Season both sides with 3/4 teaspoon kosher salt and 1/4 teaspoon black pepper.
Heat a 10-12 inch skillet over medium-high heat. Add 2 tablespoons olive oil and heat until shimmering but not smoking (approximately 350-375°F).
Place chicken breasts presentation-side down in the hot skillet. Cook undisturbed for 5-6 minutes until deeply golden brown. Flip once and cook for an additional 4-5 minutes until internal temperature reaches 160°F.
Transfer chicken to a plate and tent loosely with foil. Rest for a minimum of 5 minutes while preparing the sauce.
Reduce heat to medium. Add remaining tablespoon of olive oil to the same skillet. Add diced shallots and sauté for 2 minutes until translucent but not browned. Add sliced garlic and red pepper flakes (if using) and cook for 30 seconds until fragrant.
Add chicken stock or white wine, scraping the bottom of the pan with a wooden spoon to release fond (browned bits). Allow liquid to reduce by half, approximately 1-2 minutes.
Add halved cherry tomatoes to the skillet. Cook for 3-4 minutes until they begin to soften and release juices. Gently press some tomatoes with the back of a spoon to release more liquid.
Add balsamic vinegar and simmer for 2 minutes until slightly reduced. Season with remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Turn heat to low.
Remove skillet from heat. Add cold butter cubes, swirling pan gently until butter is melted and sauce is glossy and slightly thickened.
Stir in torn basil leaves, reserving some for garnish.
Return chicken to the skillet to rewarm briefly. Spoon sauce over chicken and garnish with remaining fresh basil.