Activate the yeast

• Pour the warm milk into a small bowl.
• Add a spoonful of the granulated sugar and sprinkle the yeast over the top.
• Stir and let rest until frothy.

Mix the dough

• Whisk the flour and salt together in a large bowl.
• Pour the yeast mixture into the bowl of a stand mixer, then add the remaining sugar, melted butter, eggs, vanilla and part of the flour mixture.
• Mix with the dough hook on low speed until barely combined, then rest.

Knead the dough

• Add the rest of the flour and mix on medium speed until the dough is elastic and smooth.
• If the dough is too sticky, add more flour a spoonful at a time, without overdoing it.

Let the dough rise

• Transfer the dough to a lightly oiled bowl and cover.
• Rise in a warm environment until doubled in size.

Cook the apple filling

• In a large saucepan over medium-high heat, add the butter, apples, brown sugar, maple syrup, salt, cinnamon and apple pie spice.
• Cook until the apples soften, stirring occasionally.
• Mix the cornstarch and water into a slurry, pour it in and stir until the mixture thickens.

Finish and cool the filling

• Remove from the heat and stir in the vanilla.
• Set aside to cool completely.

Prepare the parchment squares

• Cut out medium or small squares of parchment paper and place them on baking sheets.
• Sprinkle each square with flour.

Roll out and fill the dough

• Turn the risen dough onto a floured surface and roll it into a large rectangle.
• Spread the cooled apple filling across the entire surface and sprinkle flour over the top.
• Fold each short side of the dough to meet in the middle, enclosing the filling, and pinch the seams and edges together.

Cut and form the fritters

• Cut the dough into a grid of small pieces, then toss and gather the bits and cut again from different directions.
• Divide the dough and form it into large or small round fritters, compacting each into a ball and adding flour if needed.
• Sprinkle flour on both sides and transfer to the parchment squares.

Rest the fritters

• Loosely cover with plastic wrap and rest while the oil heats.

Heat the frying oil

• Fill a large heavy-bottomed pot with oil to about 2 inches deep.
• Heat on medium-high until it reaches 360-375 °F.
• Line a wire rack with a thick layer of paper towels.

Fry the fritters

• Carefully place one or two fritters into the hot oil, parchment side on, and after a minute pull the parchment away.
• Fry until deeply golden brown on each side.
• Remove with a slotted spoon and drain on the prepared rack. Repeat with the remaining fritters.

Make the glaze

• Brown the butter, then whisk it together with the powdered sugar, maple syrup, milk, vanilla and a pinch of salt in a large bowl.

Glaze and serve

• Dunk each fritter into the glaze.
• Set on a cooling rack until the glaze sets, then serve.