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Thinly slice the shallot and mince garlic. Set aside.
Heat olive oil in a large skillet over medium-high heat. Once the oil is hot, brown the chicken for 2 minutes on each side. Transfer the chicken to a plate and set it aside.
Add the remaining tablespoon of fig jam to skillet, stir, and cook for 1 minute. Add chicken broth, balsamic vinegar, and fresh thyme to the skillet. Stir to combine and let the mixture come to a boil.
Transfer the chicken to a plate and spoon the sauce over it or serve it straight out of the pan.