Heat the oven to 350°F. Spread the bread cubes on a baking sheet and bake until lightly toasted, 8 to 10 minutes. Remove from oven and let them cool.
Meanwhile, melt the butter in a large skillet over medium heat. Add the apples and ¼ cup of the sugar. Cook until the apples are tender and the sugar turns light caramel in color, 10 to 15 minutes. Remove the pan from the heat and stir in the cinnamon.
In a large bowl, whisk 3 cups of the milk together with the eggs, egg whites, vanilla extract, and remaining ¾ cup of sugar. Fold in the bread and apples until thoroughly coated; let the mixture stand for 10 minutes.
Coat a 2½-quart baking dish with cooking spray. Add the bread mixture and compress gently to saturate the bread. Bake until the custard is just set and the top is golden, 50 to 60 minutes.
Combine the caramels and remaining ¼ cup of milk in a 2-cup glass measure or bowl. Microwave on high until the mixture can be stirred smooth, 1 to 2 minutes, stopping every 30 seconds after 1 minute of cooking to check the consistency. Cool the caramel sauce slightly; then drizzle it over each serving.
Tip: The caramel sauce can be kept covered in the glass container in the refrigerator for up to two days. To reuse it, microwave again as directed above.
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