In a small saucepan, over medium-high heat, heat the water until it's lukewarm. Pour the yeast into the bowl of an electric mixer and pour the warm water over it. Allow it to sit for 15 to 20 minutes.
Add butter, whole egg, egg yolk, and vanilla and give it a quick stir to combine. Add bread flour, sugar, baking powder, and salt and knead with the hook until it comes together to form a smooth ball. If the dough is sticky, add flour in small increments until it comes together.
Coat the sides of the bowl with oil and cover with plastic wrap. Set aside in a warm place to rise, about 1 hour.
Turn out the dough onto a lightly floured surface. Roll and stretch it into a 10-by-15-inch rectangle, about 1/4 inch thick. Spread a little more than half of the reserved apple filling over the dough, and fold into thirds. (See picture for a visual).
Pat down and stretch the dough into another 10-by-15-inch rectangle. It will be a bit messy and there will be holes but it's okay! Spread the remaining apples over the dough, and fold into thirds again. Fold into a square and return the dough to the oiled bowl. Cover with plastic wrap and set aside for about 30 minutes or until doubled in size.
Pour oil into a shallow pot, preferably a wok, stopping at about halfway—just enough room to submerge your fritters. Set it over medium heat and clip a candy thermometer to the side. You'll want your oil to reach 375°F.
Turn out dough onto a clean surface and stretch into an even rectangle that's about 1-inch thick. Using a 2-inch round cutter, cut out circles and stretch them into misshapen ovals. Carefully lower the apple fritters into the oil. Depending on the size of your saucepan, you should be able to do 3 to 4 at a time.
When the apple fritters are a deep golden brown—after about 1 to 2 minutes—use a slotted spoon to flip them over; cook until they're a deep golden brown all over, 1 to 2 minutes more. Transfer the apple fritters to a plate lined with paper towels. Repeat with the rest of the dough.