Pat chicken dry with paper towels. Heat oil in a large nonstick skillet over medium-high heat until hot. Season chicken with thyme, salt, and pepper. Dredge in flour. Add chicken to skillet, skin side down; reduce heat to medium, and cook 16 minutes or until skin is browned. Turn and cook 5 minutes or until browned. Remove chicken from pan; keep warm. Discard pan drippings.
Melt butter in skillet over medium heat. Add figs and shallots; sauté 3 minutes or until shallots are tender. Add port; bring to a boil. Boil 1 minute; add beef broth, reduce heat, and simmer 3 minutes. Remove half of figs from skillet using a slotted spoon; keep warm with chicken. Process remaining figs and sauce in a food processor until smooth, stopping to scrape down sides as needed. Pour mixture through a wire-mesh strainer back into the skillet, discarding pulp and seeds. Return chicken and figs to sauce, and heat over medium-high heat 9 minutes or until chicken is done. Drizzle with vinegar, if desired. Cut chicken diagonally across the grain into thin strips. Serve sauce over chicken and figs. Garnish, if desired.