Warm milk to about 110°F. Stir in yeast and sugar. Let sit 5–10 minutes until foamy.
Mix in melted butter, salt, and egg. Gradually add flour until a soft dough forms.
Knead dough on a floured surface for 6–8 minutes until smooth and elastic. Place in a greased bowl, cover, and let rise 1 hour or until doubled in size.
In a bowl, mix apples with brown sugar, cinnamon, and nutmeg.
Punch down dough and roll into a large rectangle (about 14x9 inches). Spread softened butter over dough, then evenly distribute apple mixture.
Roll tightly from the long side and slice into 12 rolls. Place in a greased 9x13 baking dish.
Cover and let rise for another 30 minutes while oven preheats to 350°F (175°C).
Bake for 25–30 minutes until golden brown.
Meanwhile, beat cream cheese and butter until smooth. Add powdered sugar, vanilla, and milk until creamy.
Spread glaze over warm rolls before serving.