Combine first 9 ingredients in a medium saucepan; bring to a boil. Cover, reduce heat, and simmer 20 minutes or until figs are tender. Strain fig mixture through a colander over a bowl, reserving cooking liquid. Discard thyme sprigs, bay leaves, and rosemary sprig. Combine fig mixture, bread, cheese, 1/4 cup broth, 2 teaspoons thyme, 1/4 teaspoon salt, and 1/2 teaspoon pepper, tossing until well blended.
Preheat oven to 350°.
Place each turkey breast half between 2 sheets of heavy-duty plastic wrap; carefully pound to about 1/2-inch thickness. Remove plastic wrap. Press half of fig mixture onto each turkey breast half, leaving a 1-inch border. Roll up, starting at short end (A). Sprinkle each roll with 2 teaspoons thyme, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Wrap each in heavy-duty plastic wrap, then wrap each in aluminum foil; twist ends to seal (B). Place on a jelly-roll pan. Bake at 350° for 1 hour and 20 minutes. Remove from oven; let stand 20 minutes.
Heat a small saucepan over medium heat; coat pan with cooking spray. Add shallots; cook 3 minutes or until tender, stirring frequently. Add reserved cooking liquid, remaining 3/4 cup broth, and remaining 1/4 teaspoon salt; bring to a boil. Combine butter and flour in a small bowl; mash with fingertips to form paste. Drop butter mixture into pan by small spoonfuls, stirring constantly with a whisk. Cook 2 minutes or until thick, stirring constantly with a whisk.
Remove foil and plastic wrap from turkey. Cut each roll into 8 slices. Serve with sauce. Garnish with rosemary and thyme, if desired.