Core and thinly slice apple; coarsely chop slices. In large bowl, toss sugar, lemon juice, flour, cinnamon, nutmeg, and apples. Dust work surface with flour.
With 3-inch round cutter, cut 9 circles from dough. Gather dough scraps; roll out and cut 3 more circles. Place about 2 tablespoons apple filling and 2 pieces caramel in center of 6 circles; top each with remaining 6 circles. Seal edges of pies with fork; transfer pies to prepared pan. Repeat process with remaining pie crust to make 6 more pies.
Whisk egg; evenly brush tops of pies and sprinkle with salt. Cut 2 vents in top of each pie. Bake in 375° oven for 30 to 35 minutes or until golden brown.