Preheat oven to 350°.
To prepare the crust, place crackers in a food processor; process until fine crumbs form. Add 2 tablespoons sugar, 2 tablespoons melted butter, and egg white; pulse 8 times or until moist. Press crumb mixture into a 9-inch pie plate coated with cooking spray. Bake at 350° for 8 minutes; cool on a wire rack. Freeze 30 minutes to overnight.
To prepare filling, melt 1 tablespoon butter in a large nonstick skillet over medium heat. Add apple; cook 10 minutes or until lightly browned, stirring frequently. Add water; bring to a boil. Cover, reduce heat, and simmer 15 minutes. Stir in 1 tablespoon brown sugar, 1/4 teaspoon cinnamon, and nutmeg; cook, uncovered, 3 minutes or until liquid evaporates. Remove from pan. Cover and chill 2 hours.
Place frozen yogurt in a chilled bowl. Stir in 1 teaspoon cinnamon and 3/4 cup apple mixture. Cover and chill 30 minutes. Spread half of yogurt mixture into frozen crust (keep remaining yogurt mixture in freezer). Top the yogurt mixture with the remaining apple mixture. Drizzle with 1/4 cup caramel topping. Cover and freeze 1 hour. Carefully spread reserved yogurt mixture over caramel. Top with graham cracker crumbs and pecans. Drizzle with 2 tablespoons caramel topping. Freeze overnight or until solid. Place pie in the refrigerator for 30 minutes before serving.