Preheat oven to 350º.
To prepare chicken, bring port to a boil in a medium saucepan. Add figs; remove from heat. Let stand, covered, 30 minutes or until soft.
Combine flour, 1/2 teaspoon salt, and 1/4 teaspoon pepper in a large zip-top plastic bag; add chicken. Seal and shake to coat. Heat 1 tablespoon oil in a large Dutch oven over medium-high heat. Add half of chicken to pan, and cook 5 minutes on each side or until lightly browned. Remove chicken from pan. Repeat procedure with 1 tablespoon oil and remaining chicken. Return chicken to pan. Add fig mixture, 1/4 teaspoon salt, 1/4 teaspoon pepper, shallots, and next 4 ingredients (shallots through 1 can broth). Cover and bake at 350º for 25 minutes. Stir in honey. Uncover and bake at 350° for 25 minutes. Stir in vinegar; place mixture in a large bowl. Cover and keep warm.
To prepare escarole, heat 1 tablespoon oil in pan over medium heat. Add escarole; cook 5 minutes or until escarole begins to wilt. Add 1/2 cup broth. Cover; cook 5 minutes or until tender. Stir in 1/4 teaspoon salt and 1/4 teaspoon pepper. Serve chicken mixture over escarole.