Preheat the oven to 325ºF. If your bread is not already in pieces, tear or cut the bread into one-inch pieces.
Spread the bread pieces out on a baking sheet and bake in the preheated oven for about 10 minutes to slightly dry the bread. After baking, let the bread cool for a few minutes to allow the remaining steam to evaporate.
While the bread is drying in the oven, prepare the apple cinnamon custard. In a medium bowl, whisk together the milk, eggs, 1 cup of applesauce, vanilla extract, sugar, and cinnamon.
Transfer the par-baked bread to a large bowl, add the raisins, then pour the apple cinnamon custard over top. Stir to combine and allow the bread to soak for at least 15 minutes to absorb all of the custard.
Meanwhile, use 1 teaspoon of the butter to grease the inside of a two-quart casserole dish. Transfer the soaked bread to the casserole dish and spread it out evenly. Cut the remaining butter into chunks and sprinkle it over the bread pudding.
Bake the bread pudding for 45 minutes, or until the internal temperature reaches 160ºF.
After removing the bread pudding from the oven, add the remaining ½ cup applesauce in dollops on top of the bread pudding. Serve warm.