In the bowl of a stand mixer fitted with the paddle attachment, stir together 2 cups of the flour, the yeast, sugar, and salt.
Add the warm milk, vanilla, and egg yolks.
Mix until smooth.
Add the remaining flour and the butter and mix until incorporated.
Continue to mix on medium speed until the dough is soft and smooth (it will be slightly sticky).
Cover the bowl with plastic wrap or damp towel and set in a draft-free warm place to rise for 1 hour, or refrigerate for up to 12 hours.
Melt the butter in a skillet over medium-high heat.
Add the apples to the pan and sprinkle with the sugar, cinnamon, and a pinch of salt.
Cook, stirring, until tender and the liquid becomes a syrupy glaze, about 10 minutes.
Set aside and let cool completely.
Roll the dough out to a 1/2-inch thickness.
Spread the apple mixture onto half of the dough.
Then fold the other half over the apples.
Using a large knife or a bench scraper, cut the dough into 1/2-inch strips.
Now cut the strips into 1/2-inch pieces in a cross pattern.
Scoop up the pieces and rearrange, cutting them again in a cross pattern (make sure the apples are well dispersed throughout the dough).
Flour your hands really well and form the dough into a log measuring 12 inches long and about 3 inches wide.
Sprinkle the log with flour and slice it crosswise into 12 pieces. Coat 2 rimmed baking sheets with flour.
Transfer the uncooked apple fritters to the baking sheets, smashing the pieces of dough and apple together.
Tuck in any apples that stray, and press each fritter into a flat round (make sure you really press the pieces together and flatten before you let them rise, or the fritters will come apart during frying).
Cover with plastic wrap and set in a draft-free warm place to rise for 25-30 minutes.
Fill a large, heavy-bottomed pot with at least 2 inches of oil (the oil should not come more than halfway up the pan).
Heat over medium-high heat until a deep-frying thermometer registers 350°F.
Using a spatula, carefully and gently lower 1 or 2 fritters into the oil; don’t crowd them.
Fry for 1 to 2 minutes per side, turning a few times, until the fritters are light golden brown and cooked through.
Remove with a spider, drain on a wire rack over a paper towel, and let cool slightly before glazing.
Repeat with the remaining fritters.
While the fritters are warm, dip the rounded side into the maple glaze, letting the excess drain back into bowl.
Place back on the rack to set.
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