1. To prepare crust, weigh or lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, 1 tablespoon sugar, and 1/2 teaspoon salt in a food processor; pulse until combined. Add 6 tablespoons chilled butter; pulse until mixture resembles coarse meal. Add 2 tablespoons ice water, and pulse until mixture forms clumps. Gently press dough into a 6-inch circle on heavy-duty plastic wrap; cover and freeze 30 minutes.
2. To prepare filling, combine apples, juice, and 1/4 teaspoon salt in a large bowl, tossing to coat. Melt 2 tablespoons butter in a 9 1/2-inch cast-iron skillet over medium-high heat. Add 3/4 cup sugar to pan; cook 4 minutes or until golden brown, stirring constantly. Remove pan from heat. Arrange half of apples, rounded side down, in a circular pattern over sugar mixture in pan. Top with remaining apples, rounded side up. Cook over medium heat 15 minutes. Remove from heat; let stand 15 minutes.
3. Preheat oven to 400°.
4. Working quickly, roll dough into an 11-inch circle on a heavily floured surface. Place dough over apples; fold edges under. Cut 4 (1-inch) slits into top of pastry using a sharp knife. Bake at 400° for 40 minutes or until crust is lightly browned. Remove from oven; let stand 5 minutes. Place a plate upside down on top of pan. Carefully invert tart onto plate. Serve warm.