Melt the butter in a skillet over medium heat. Add the apple, brown sugar and water. Saute for about 10 minutes, or until the apples have cooked down and the liquid has mostly evaporated. Let it cool.
In a small bowl, whisk together the eggs, sugar, vanilla and milk. Set it aside.
Add the rest of the ingredients to a large bowl and mix well.
Add the wet mixture to the dry mixture and stir until its mostly incorporated. Then add the cooked apples and continue mixing until it everything is combined. The dough should be a little softer than cookie dough. Add a little more flour if it seems too wet, and a little more milk if it seems too dry.
Heat the oil in a pot or wok. The oil should be between 310º F and 320º F. If the oil is too cool, it will make the andagi very oily, and if it’s too hot, it will burn before the inside is cooked.
Using spoons or a small ice cream scoop, drop the mixture into the oil (about a tablespoon of dough). They will puff up while cooking, so don’t over crowd the pan. Cook the andagi for about 8 minutes, or until golden brown and cooked in the middle (make sure to flip them so both sides cook evenly). Test to see if the andagi is done by poking them with a skewer. It should come out clean. Remove them from the oil and drain them on a paper towel. Roll the andagi in cinnamon sugar (optional) and enjoy! The andagi is best served warm, but will stay fresh at room temperature for a few days.