Mix goat cheese, thyme, and garlic until blended. Place 1 rounded tablespoon goat cheese mixture into center of each pear half. Puree chicken broth, figs, vinegar, and sugar in batches in a blender until smooth. Pour into strainer set over a bowl. Press liquid (sauce) into the bowl and discard solids. Pour half of the fig sauce into 13 x 9 x 2-inch pan.

Place chicken breasts over fig sauce in the pan. Sprinkle with salt and pepper. Place a filled pear half cut side down over each chicken breast and drizzle with remaining fig sauce. Cover pan with foil and bake in conventional oven at 375 degrees for 20 minutes. Remove foil and continue to bake, basting occasionally, 30 to 40 minutes until chicken breasts reach an internal temperature of 160 degrees. Cut chicken breast and pear in half to serve. Spoon sauce over pear. Garnish with cut fig and thyme sprig.