Step 1: In the bowl of a stand mixer fitted with the dough hook, combine warm milk, active dry yeast, and a pinch of sugar. Let sit for 5-10 minutes until foamy.
Step 2: Add flour, granulated sugar, salt, and eggs to the yeast mixture. Mix on low speed until a shaggy dough forms, then increase to medium speed and knead for 5 minutes.
Step 3: With the mixer on medium-low, add softened butter one tablespoon at a time, ensuring each addition is incorporated. Knead for 10-15 minutes until smooth and elastic. Transfer dough to an oiled bowl, cover, and let rise for 1-2 hours or overnight in the fridge.
Step 4: For the filling, combine diced apples, granulated sugar, brown sugar, cinnamon, nutmeg, allspice (if using), lemon juice, and water in a saucepan. Cook over medium heat, stirring, until apples soften (8-10 minutes).
Step 5: Whisk cornstarch with 1 tablespoon cold water to make a slurry. Stir into apple mixture and cook for 1-2 minutes until thickened. Remove from heat and cool completely.
Step 6: Gently punch down risen brioche dough. Roll on a lightly floured surface to ½-inch thickness. Cut out donut rounds with a 3-inch cutter. Place on parchment-lined baking sheets, cover, and let rise for 30-45 minutes.
Step 7: Heat 2-3 inches of vegetable oil to 350°F (175°C) in a large pot. Fry donuts 2-3 at a time for 1-2 minutes per side until golden brown. Place on a wire rack to drain.
Step 8: While warm, roll donuts in granulated sugar or cinnamon-sugar mixture.
Step 9: Once slightly cooled, use a small knife or piping bag with a thin nozzle to create a hole on the side of each donut. Pipe the cooled apple pie filling into each donut. Serve immediately.