Stir together the flour, baking powder, baking soda, cinnamon, salt, brown sugar, and white sugar.
Add the apples, eggs, oil, and beer, stir until combined.
Heat the oil in a pot over medium-high heat. Clip a deep-fry thermometer on the side, adjust the heat to maintain 350°F.
Drop a spoonful of dough into the hot oil. Cook on both sides until golden brown and cooked through. About 2 minutes per side. Remove with a metal slotted spoon.
Allow to drain and dry on a stack of paper towels.
Stir together the glaze ingredients, thin with water until desired consistency.
Drizzle the glaze over the doughnuts.