In the bowl of a stand mixer fitted with a dough hook, combine the flour, granulated sugar, instant yeast, and salt. Add the warm milk, 1/4 cup melted butter, and egg. Mix on low speed for 2 minutes until a shaggy dough forms, then increase to medium speed and knead for 5-6 minutes until the dough is smooth, elastic, and pulls away from the sides of the bowl.
Transfer the dough to a lightly oiled bowl, turning to coat. Cover the bowl with a clean kitchen towel and let the dough rise in a warm place for 1 hour, or until it has doubled in size.
While the dough rises, make the apple filling. In a medium bowl, stir together the chopped apples, brown sugar, cinnamon, and nutmeg until the apples are evenly coated.
Punch down the risen dough and turn it out onto a lightly floured surface. Roll the dough into a 12x16-inch rectangle. Spread the 2 tablespoons of softened butter evenly over the surface, then sprinkle the apple mixture evenly over the butter, leaving a 1-inch border on one long side.
Starting from the long side opposite the border, tightly roll the dough into a log. Use a sharp serrated knife to slice the log into 8 equal rolls. Place the rolls in a greased 9-inch round cake pan. Cover the pan and let the rolls rise for 30 minutes until puffy.
Preheat your oven to 350°F. Bake the rolls for 25-30 minutes, or until they are golden brown on top and sound hollow when tapped.
While the rolls bake, whisk together the powdered sugar, 2 tablespoons milk, and vanilla extract in a small bowl until you have a smooth, pourable glaze. Drizzle the glaze over the warm rolls before serving.