Add vegetable oil to a large pot until it is about 2 inches deep. Turn the heat to medium and let the oil heat up while you mix up the batter.
Peel and core both apples, then cut into small 1/4-inch cubes. Set aside.
Add the dry ingredients to a large bowl: all-purpose flour, brown sugar, baking powder, cinnamon, salt. Stir until combined.
In a separate small bowl, combine the wet ingredients: milk, eggs, and vanilla. Whisk until well combined.
Pour the wet ingredients into the dry ingredients and mix just until there are no streaks of flour remaining. Add the apple chunks into the batter and fold until they're just combined in.
When the oil reaches 350F, use two spoons to scoop up about 1/4 cup of batter. Carefully drop the batter into the oil. Flip it every 2 minutes until it is deeply golden brown on both sides. Carefully remove from the hot oil and transfer to a cooling rack. Repeat with the remaining batter, only cooking 3-4 donuts at once so the pot doesn't get too crowded.
Once the donuts are all cooked, make the glaze. Add the powdered sugar, maple syrup, vanilla extract, and salt to a bowl. Add 1 tablespoon of milk and whisk to combine. If the icing is really thick, add 1 tablespoon additions of milk until the icing is your desired consistency.
While the donuts are still warm, dip each one in the icing so both sides are covered. Then place them back on the cooling rack for the icing to set up. These donuts are best enjoyed warm so dive in!