Melt the 1 tablespoon butter in a skillet over medium heat, then add the 2 diced apples, 2 tablespoons brown sugar, and 1 teaspoon cinnamon. Cook for 4 to 5 minutes, until softened and syrupy. Spread them on a plate and let them cool completely, since warm apples would melt the dough.
In a large bowl, whisk together the 3½ cups flour, ¼ cup granulated sugar, 2¼ teaspoons instant yeast, 1 teaspoon salt, and 1 teaspoon cinnamon. Add the ¾ cup warm milk, ¼ cup melted butter, and 2 eggs, then mix until a soft dough forms.
Knead for about 5 minutes, until smooth and elastic. Place the dough in a greased bowl, cover, and let it rise in a warm spot for 1 hour, or until doubled.
Punch down the dough and roll it out on a floured surface into a large rectangle, about ½ inch thick. Scatter the cooled cinnamon apples evenly over the surface.
Fold the dough over the apples like a letter, then roll it out again. Fold and roll one more time. This folding is what gives you those craggy layers and pockets of apple, so do not skip it. The dough will look rough and messy, which is exactly right.
Roll the dough to about ¾ inch thick and cut it into 10 large pieces. They do not need to be neat, ragged edges are the whole look.
Place the fritters on a floured tray, cover loosely, and let them rise for 30 minutes, until puffy.
Pour 2 inches of vegetable oil into a heavy pot and heat it to 350°F (175°C). Getting the temperature right is the whole game, too cool and they go greasy, too hot and they burn before the inside cooks.
Working in batches, fry the fritters for 2 to 3 minutes per side, until deep golden brown and cooked through. Drain on a wire rack set over paper towels and let them cool for about 10 minutes.
Make the glaze. Whisk together the 2 cups powdered sugar, 3 tablespoons milk, and ½ teaspoon vanilla until smooth. Dip the top of each fritter into the glaze, or dunk them fully, then set them back on the rack to let the glaze set.