Sprinkle both sides of chicken evenly with 1 1/2 teaspoons thyme, 1/4 teaspoon salt, and pepper. Heat oil in a large nonstick skillet over medium-high heat. Add chicken; cook 6 minutes on each side or until done. Remove from pan; keep warm.

Reduce heat to medium; add butter to pan. Add onion; sauté 3 minutes. Add broth, vinegar, soy sauce, and figs. Simmer until sauce is reduced to 1 cup (about 3 minutes). Add 1 tablespoon thyme and 1/4 teaspoon salt. Cut chicken breast halves lengthwise on the diagonal into slices. Serve sauce over chicken.