Preheat oven to 350°.
Lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, brown sugar, and salt, stirring well with a whisk. Cut in butter with a pastry blender or 2 knives until mixture resembles coarse meal. Press mixture firmly into a 13 x 9-inch baking dish coated with cooking spray.
Combine figs, water, and 1/4 cup granulated sugar in a medium saucepan; bring to a boil over medium-high heat. Cook 5 minutes or until figs are tender and sugar dissolves. Cool slightly. Place fig mixture and juice in a blender; process until smooth. Gently spread fig mixture over prepared crust.
Place remaining 1/4 cup granulated sugar, cheese, vanilla, and egg in a medium bowl; beat with a mixer at medium speed until smooth. Pour over fig mixture; spread to edges. Bake at 350° for 30 minutes or until set and lightly browned. Cool in pan on a wire rack; sprinkle with powdered sugar.