Prep the apples
Core, peel, and chop apples into small pieces. Submerge them in cold water with 2 teaspoons of lemon juice to prevent them from browning. Push the apples down and place a piece of parchment paper on top so the apples don’t bob above the surface of the water. Set aside or refrigerate (for up to 24 hours) until ready to use.
Make the dough
Whisk together the yeast, 3 cups of the flour, sugar, salt, and spices in the bowl of a stand mixer. With a dough hook and the mixer on low speed, mix in the milk, followed by the eggs. Continue mixing until the dough gathers into a ball around dough hook, 2 to 4 minutes.
Continue mixing the dough, and add in the butter one tablespoon at time. Allow each tablespoon of butter to combine into the dough before adding another. If dough starts to fall away from the dough hook, add the remaining 1/4 cup of flour 1 tablespoon at a time until dough comes back together before adding in next butter piece.
Transfer the dough to a lightly oiled bowl, turn to coat, and cover with a clean kitchen towel. Let the dough rise in a warm spot until doubled in size, 1 to 2 hours.
Make the apple filling while the dough rises
Heat the butter for the filling in a skillet over medium-high heat until butter turns brown and smells nutty. Drain the apples and add them to the pan with the butter. Sprinkle with sugar and spices, and cook until tender. Add in vinegar and cook until liquid is reduced by half. Set aside to cool.
Assemble the fritters
Turn the dough out onto a floured surface. Using a rolling pin, roll the dough into a 1/2-inch-thick rectangle. Starting from the end closest to your body, scatter 3/4 of the apples over the surface of the dough, then roll it up in a long tube.
Now rotate the tube of dough so it’s perpendicular to your body. Flatten the tube slightly, then scatter the remaining apples over the surface. Roll up the tube like a snail’s shell — the finished dough will look like a large ball.
Using a rolling pin, roll out the apple-filled dough ball to a 1/2-thick rectangle. Some apples may pop out — don’t worry if they do, just set them aside and pop them back in where you can. Use a bench scraper to cut the dough into 2 1/2-inch squares to make 12 fritters. Use your hand to and pull the four corners of each square towards the center. Again, don’t worry about any apples falling out or if they are exposed. Pinch dough together if necessary to create a rough circle.
Place the shaped fritters on a Tovolo Baking Mat lined baking sheet and loosely cover with plastic wrap. Let rise for about 30 minutes or until the fritters puff and almost double in size
Make the glaze
Sift the powdered sugar into a medium bowl. Stir in the milk and vanilla until smooth. Set aside; warm in the microwave in 10-second bursts if the glaze starts to harden.
Deep-fry the fritters
Heat 6 to 7 inches of oil to 360°F in a large Dutch oven. Meanwhile, line a wire rack with paper towels.
Carefully lower 2 or 3 fritters into the hot oil. Fritters will float and not sink. Fry on one side for 30 to 60 seconds or until golden brown, then flip the fritter and fry the other side for 30 to 60 seconds or until golden brown. Remove fritter with a spider or slotted spoon and drain on prepared wire rack. Repeat with the remaining fritters. (Between batches, make sure the oil temperature returns to 360°F and remove any floating pieces of dough or apples.) **My fritters took more than 60 seconds on each side.
Let fritters cool slightly, then brush the glaze on top. Serve immediately.