Preheat oven to 400°.
Lightly spoon flour into dry measuring cups; level with a knife. Place flour in a medium bowl, and cut in butter with a pastry blender or 2 knives until mixture resembles coarse meal. Combine 2 tablespoons water and egg, stirring with a whisk. Add egg mixture to flour mixture, stirring just until moist. Turn dough out onto a large piece of heavy-duty plastic wrap; knead lightly 5 times (dough will be sticky). Pat dough into a disk. Cover with additional plastic wrap, and chill 30 minutes.
Combine 4 tablespoons water and sugar in a 9-inch cast-iron skillet over medium-high heat. Cook 10 minutes or until golden, stirring only until sugar dissolves. Remove from heat; gently stir in small circles to evenly distribute cooked sugar. Let stand 5 minutes.
Arrange apple quarters tightly in a circular pattern over sugar in pan, beginning at the outside edge. Cut 2 apple quarters in half, and arrange, points up, in center of pan. Place pan over medium heat; cook 20 minutes (do not stir), pressing apples slightly to extract juices. Remove from heat; let stand 10 minutes. Sprinkle cinnamon over apples.
Remove plastic wrap covering dough. Turn dough onto a lightly floured surface; roll dough into an 11-inch circle. Place over apple mixture, fitting dough between apples and skillet. Bake at 400° for 20 minutes or until lightly browned. Let cool 10 minutes. Invert tart onto a plate. Serve with crème fraîche.