Stir together first 6 ingredients in a small saucepan. Add 1 cup chicken broth. Bring to a boil; cover, reduce heat, and simmer 10 minutes. Cool 5 minutes. Process mixture in a food processor 1 minute or until pureed. Set aside.
Truss each quail with heavy string or by inserting a wooden pick through end of one leg, tip of breast, and end of other leg. Sprinkle quail with 1 teaspoon salt and 1/2 teaspoon pepper.
Heat oil in a large skillet over medium-high heat until hot. Fry quail, in batches, 5 minutes on each side or until browned. Place quail, breast side up, on a baking sheet or roasting pan. Brush with 1/3 cup fig puree.
Bake, uncovered, at 475° for 15 to 16 minutes or until golden. Arrange quail on a platter, remove string or picks, and cover to keep warm.
Combine remaining fig puree, 2 cups broth, 1 teaspoon salt and 1/4 teaspoon pepper in a saucepan. Bring to a boil; reduce heat and simmer, uncovered, 1 minute. Pour mixture through a wire-mesh strainer into a bowl, discarding solids. Return sauce to saucepan. Combine cornstarch and water, stirring until smooth. Stir into sauce; cook over medium heat until sauce thickens. Serve sauce with quail.
*If you can't find quail, use 6 large skinless, boneless chicken-breast halves. Sprinkle with 1 teaspoon salt and 1/2 teaspoon pepper. Heat oil in a large skillet as directed above. Brown chicken in hot oil on both sides. Place chicken on a baking sheet; brush with 1/3 cup fig puree. Bake, uncovered, at 475° for 15 to 20 minutes or until chicken is done.