Preheat the oven to 375 degrees.
In a small bowl, mix the water, arrowroot, sugar, lemon juice, spices, salt, and vanilla.
Toss the mixture with the apple slices and pour them into an 8 x 8 inch baking dish.
To make the cobbler topping, mix the flour, salt, baking powder, and cinnamon in a small bowl.
Cut in the oil until the mix is pebbly in appearance.
Lightly beat the egg (if using) with the milk.
Add it (or 1/2 cup of milk total if not using the egg) to the flour.
Mix until all dry ingredients are moistened.
Dollop the batter evenly atop the apples (see picture below).
Bake the cobbler for 30 minutes.
Cool, cut, and enjoy!