Preheat oven to 350 degrees F (175 degrees C). Grease 2 baking sheets.
Beat applesauce, white sugar, brown sugar, and vanilla in a large bowl. In another bowl, use an electric mixer to beat egg whites until they are frothy and begin to firm up. Fold egg whites into applesauce mixture. Combine the flour, baking soda, salt, and cinnamon. Fold into the egg mixture. Stir in the oats, chocolate chips, walnuts, and cranberries. Drop by heaping teaspoons on prepared baking sheets.
Bake cookies in preheated oven until set and lightly browned, about 10 minutes. Remove immediately to wire racks to cool.