In a bowl, whisk together warm milk, sugar, and yeast. Stir until the yeast is dissolved. (If using active dry yeast, let it sit for about 5 minutes until foamy.)
Add the egg and egg yolk and mix well.
Add the flour and salt, then mix until a shaggy dough forms.
Add the butter and knead for 8–10 minutes, until smooth, elastic, and it passes the windowpane test.
Place the dough in a lightly greased bowl, cover, and let rise for 1- 1 ½ hours, or until doubled in size.
Melt the butter in a saucepan. Add the apples, sugar, cinnamon, nutmeg, salt, and lemon juice. Cook over medium-low heat for 6-7 minutes, until the apples are tender.
Mix the cornstarch with water, add to the apples, and stir until thickened.
Remove from the heat, stir in the vanilla, and let cool completely. Transfer to a piping bag.
Roll the dough into a large rectangle and cut into 10 equal bars.
Place the bars on parchment paper, cover, and let proof for 30-45 minutes.
Heat the oil to 350°F (180°C). Fry the donuts for 1-2 minutes per side, until golden brown. Let cool slightly, then poke a hole in the center using a skewer or chopstick.
Mix all the glaze ingredients until smooth.
Fill each donut with the apple pie filling, then dip in the maple glaze. Place on a wire rack and let the glaze set. Enjoy!
