Preheat oven to 375 degrees. Butter an 8-inch square baking dish. On a rimmed baking sheet, arrange bread in a single layer and bake until dry, about 12 minutes. Transfer half the bread to a food processor and pulse until fine crumbs form. In a medium saucepan, combine butter, 1/4 cup sugar, apple juice, and salt. Cook over medium until sugar dissolves, 2 minutes. Remove from heat and stir in bread pieces and crumbs; let stand 10 minutes.
In a medium bowl, toss apples with 1/2 cup sugar, rum, lemon juice, and flour. Transfer to dish; top with bread mixture. Cover with foil and bake until apples are tender, 40 to 45 minutes. Uncover and bake until top is golden, about 15 minutes. Let cool 5 minutes before serving.