Pat the chicken breasts dry with paper towels. If they are unevenly thick, place them between sheets of plastic wrap and pound to an even 1/2-inch thickness. Season both sides with salt and pepper.
Heat the olive oil in a large skillet over medium-high heat. Once the oil is shimmering, carefully place the chicken in the pan.
Cook the chicken for 5-7 minutes on each side, until it's golden brown and cooked through. The chicken is done when a meat thermometer inserted into the thickest part reads 165°F (74°C).
Transfer the cooked chicken to a clean plate and let it rest while you make the sauce.
Reduce the heat under the skillet to medium-low. Add the minced garlic and fresh thyme, and cook for about 30 seconds until fragrant, stirring continuously.
Pour in the white wine to deglaze the pan, using a wooden spoon to scrape up any browned bits stuck to the bottom.
Stir in the chicken broth, fig preserves, and balsamic vinegar. Whisk until the preserves have dissolved into the sauce.
Bring the sauce to a gentle simmer and let it cook for 2-3 minutes, or until it has thickened slightly and can coat the back of a spoon.
Slice the rested chicken and serve with the warm balsamic fig sauce spooned over the top.