Coat a nine-inch glass baking dish with a thin film of butter.
Peel and core the apples, and cut into dice about 3/4 inch on a side. Place the apples into the baking dish and toss with the cranberry or orange juice to coat. Add the brown sugar, cinnamon, mace, and cane syrup, and toss the apples again.
Cream the butter and sugar together with an electric mixer in a bowl, until the mixture is fluffy and not gritty. Mix in the egg and the egg white until they disappear completely.
Add about 1/2 cup of flour and mix in completely. Add about 1/4 cup of milk and mix, and continue alternating flour and milk, with one final addition of flour at the end.
With a rubber spatula, spread this batter over the apples, pushing it down to fill the spaces between the apples on top, but not at the bottom.
Put the dish into the center of the oven at 350 degrees and bake for 45-50 minutes, until lightly browned. (I recommend using the convection feature of your oven, if you have it.)
Let the pandowdy cool for fifteen minutes. Turn it upside down on top of a serving plate, or just set the baking dish out to be scooped from. Spoon warm sauce anglaise over it at the table.