In a microwave safe dish, combine the milk and the butter and heat for 1-2 minutes until the butter is melted and the milk is about 100°F (warm to the touch but not hot). Whisk in the egg and set aside.
In the bowl of a stand mixer (or in a large bowl with a wooden spoon or dough whisk) stir together 3 cups (15oz) of the flour, yeast, sugar, salt and cinnamon.
Add the milk mixture and mix with the dough hook for 30-60 seconds until the dough clears the sides of the bowl. If the dough is still really wet and isn’t clearing the sides of the bowl, sprinkle in a little more flour, a tablespoon at a time, until it does. Knead the dough for 5 minutes by machine, or 10 by hand until the dough is soft and smooth.
Turn the dough out onto a lightly floured surface and use your hands to mix in the apple pieces. It will take a few minutes of kneading by hand to incorporate all the apples, and the moisture will make the dough stickier, so sprinkle in more flour as needed.
Transfer the dough to a lightly oiled bowl and turn it to coat.
Cover the bowl tightly with plastic wrap and let the dough rise for 2 hours, or until doubled.
Punch down the dough and turn it out onto a lightly floured surface.
Cut the dough into either 12 or 24 equal sized pieces (depending on whether you would like large or small fritters). Line two baking sheets with parchment paper or silicone mats.
Working with one piece of dough at a time, use kitchen shears to cut the dough into 10-12 smaller pieces, making a round pile with the pieces on the prepared baking sheet.
Repeat with the rest of the dough until you have all the apple fritters shaped on the baking sheets.
Cover loosely with plastic wrap and let rise for another 60-90 minutes.
Preheat the oven to 350°F. Bake the fritters, one sheet at a time, for 14-15 minutes (only 12-13 minutes if you’re making the smaller ones) or until golden brown.
Transfer to a wire rack to cool for 5 minutes.
In a medium bowl whisk together the powdered sugar, milk, and vanilla extract until smooth.
Drizzle the glaze over top of the warm apple fritters.
With yeast dough, it’s best to go by weight rather than a cup measurement for the flour, so if you have a kitchen scale, I highly recommend using it. The dough can be affected by a number of factors, so start with the lower amount of flour and add more if needed as adding too much flour will make the fritters too dense.