In a mixing bowl, combine the warm milk, sugar, and yeast. Let sit for 5-10 minutes until foamy.
Add the melted butter, egg, and salt, then gradually mix in the flour until a dough forms.
Knead the dough for 5-7 minutes, then cover and let rise in a warm place for 1 hour or until doubled in size.
In a small bowl, mix brown sugar and cinnamon for the filling.
Roll out the dough into a 15x9-inch rectangle and spread softened butter over the surface.
Sprinkle the cinnamon sugar mixture evenly over the buttered dough, then add the diced apples.
Roll the dough tightly into a log and cut into 12 equal rolls.
Place the rolls in a greased baking dish, cover, and let rise for another 30 minutes.
Preheat the oven to 350°F (175°C) and bake for 22-25 minutes, or until golden brown.
In a small bowl, whisk together powdered sugar, milk, and vanilla to make the glaze.
Drizzle the glaze over the warm cinnamon rolls before serving.