Mix the sweetened condensed milk with applesauce in a saucepan, and thin with apple juice. Whisk in cinnamon; bring to a boil. Beat the eggs in a bowl; whisk 1 tablespoon of the hot applesauce mixture into the eggs; whisk 1 more tablespoon of applesauce mixture into the eggs, and repeat, mixing in 1 tablespoon at a time, until about 1/3 of the applesauce mixture is whisked into the eggs. Return the egg mixture to the saucepan with the remaining applesauce mixture, and whisk until smooth. Pour the filling into the crust.
Bake until the filling is set, about 1 hour; allow to cool before serving. Refrigerate leftovers.