Pour the oil into a heavy, deep saucepan or Dutch oven. It should be at least 1 ½ inches deep, so add a little more if your pot is wide. Clip on a deep-fry thermometer and heat the oil over medium heat to 365°F.

While the oil heats, whisk the milk and the slightly beaten egg together in a medium bowl. Stir in the chopped apples.

In a separate bowl, whisk together the flour, 1 tablespoon sugar, baking powder, ¾ teaspoon cinnamon and salt.

Add the flour mixture to the egg mixture and stir just until all the dry ingredients are moistened. The batter should be thick and hold its shape on a spoon. If it's runny, stir in more flour, 1 tablespoon at a time.

Carefully drop the batter into the hot oil by heaping tablespoons (about 2 tablespoons each), 3 or 4 at a time so the pot isn't crowded.

Fry for about 4 minutes, turning once or twice, until the fritters are deep golden brown on both sides and cooked through in the middle.

Lift the fritters out with a slotted spoon and set them on a wire rack over paper towels to drain. Let the oil come back up to 365°F before you fry the next batch.

Whisk the powdered sugar, 2 tablespoons milk, vanilla and ¼ teaspoon cinnamon (if using) until smooth. It should be thick but pourable. Add more milk, 1 teaspoon at a time, if it's too thick.

Dip the tops of the warm fritters into the glaze, or drizzle it over them, then let the glaze set for about 10 minutes before serving.