Combine flour, yeast, granulated sugar, and salt in a stand mixer bowl. Add warm milk, 3 tbsp melted butter, and egg. Knead with dough hook on medium for 5 to 7 minutes until smooth. Cover and rest 10 minutes.

In a skillet over medium heat, cook 2 tbsp melted butter, diced apple, brown sugar, and cinnamon for 4 to 5 minutes until apples soften. Let cool.

Roll dough into a 15x12-inch rectangle. Brush with remaining 2 tbsp melted butter, spread cooled apple mixture evenly, leaving a 1-inch border at the top edge.

Roll up tightly from the long side, pinch seam, slice into 12 pieces. Place in greased 9x13-inch dish, cover, and let rise 30 minutes. Preheat oven to 375°F (190°C) toward the end of the rise.

Bake 22 to 25 minutes until golden and a toothpick inserted in dough comes out clean. Cool 5 minutes.

Beat cream cheese, powdered sugar, vanilla, and 2 tbsp milk until smooth. Thin with extra milk if needed, then spread over warm rolls and serve.