Place 8 wonton skins on a work surface. Fill center of each with 1/2 teaspoon fig mixture and 1/2 teaspoon goat cheese. Brush edges with egg, place another wonton skin on top, and press edges together lightly. With a cookie cutter, cut out a 2-inch round. Press down around filled area of wonton. Repeat with remaining wonton skins and fillings.
In a medium saucepan, heat oil to 375°. Add wontons, 3 or 4 at a time, and fry until golden brown on each side. Remove from oil and drain on paper towels..