Whisk the warm milk, yeast, and 1 Tablespoon of sugar together in the bowl of your stand mixer fitted with a dough hook or paddle attachment. Cover and allow to sit for 5 minutes.

Add the remaining sugar, butter, eggs, vanilla extract, salt, and 1 cup of flour. Beat on low speed for 30 seconds, scrape down the sides of the bowl with a silicone spatula, then add the remaining flour. Beat on medium speed until the dough comes together and pulls away from the sides of the bowl, about 2 minutes. If needed, add more flour, 1 Tablespoon at a time, until the dough pulls away from the sides of the bowl.

Keep the dough in the mixer and beat for an additional 5–7 full minutes, or knead by hand on a lightly floured surface for 5–7 full minutes.

Lightly grease a large bowl with oil or nonstick spray. Place the dough in the bowl, turning it to coat all sides in the oil. Cover the bowl with aluminum foil, plastic wrap, or a clean kitchen towel. Let the dough rise in a relatively warm environment for 1–2 hours or until double in size.

In a small bowl, whisk the cornstarch, water, and vanilla until the cornstarch is dissolved. Set aside. In a medium saucepan over medium heat, combine the apples, brown sugar, cinnamon, and nutmeg. Cook, stirring, until the mixture comes to a boil, about 5 minutes. Stir in the cornstarch slurry. Cook for another minute. Remove from heat, transfer to a heat-proof bowl, and cool completely.

Cut parchment paper into 12 squares, about 5–6 inches each. When the dough is ready, punch it down to release the air. In a small bowl, mix the flour, cinnamon, and nutmeg together.

Sprinkle some of the spiced flour over a large work surface. Turn out the dough and, with lightly floured hands or a rolling pin, press or roll it into a large oval about ½ inch thick. Spoon the cooled apple filling over one half of the dough, then sprinkle some of the spiced flour over the filling. Fold the unfilled half of the dough over the apples and seal the edges.

Use your hands to gently shape the filled dough into a long oval, making sure the edges are sealed. Using a sharp knife or bench scraper, make 11 cuts across the filled dough, cutting mostly but not quite all the way through. Then make 3–4 perpendicular cuts to create a crosshatch pattern. Sprinkle on more spiced flour as needed to help absorb any exposed sticky filling. Now divide the dough into 12 roughly equal portions.

Place each fritter on a parchment square and set it in a standard 12-cup muffin pan. Loosely cover and let the fritters rest while you heat the oil.

Pour the oil into a large heavy-bottomed pot set over medium heat and attach a candy/oil thermometer to the pot. Heat the oil to 350°F (177°C).

When the oil is hot enough, carefully lower 2–3 fritters, still on their parchment squares, into the oil. Fry for 2 minutes. Using tongs, flip the fritters and remove and discard the parchment. Fry for another 1–2 minutes, flipping once more if needed, until deeply golden brown on both sides.

In a medium bowl, whisk together confectioners’ sugar, milk, and vanilla extract until smooth. If needed, add more milk to thin out or more confectioners’ sugar to thicken.

Dunk one side of each fritter into the glaze. Return to the rack and let the excess glaze drip off. Allow the glaze to set.