Prepare the dough: In a small bowl, combine warm milk, yeast, and granulated sugar. Stir and let sit for 5-10 minutes until frothy.

1. In a large bowl, combine melted butter, salt, and egg. Add yeast mixture and stir to combine.
2. Gradually add flour, one cup at a time, mixing until a dough forms. Knead on a floured surface for 5-7 minutes until smooth and elastic.
3. Place dough in a greased bowl, cover with a towel, and let rise in a warm place for 1-1.5 hours until doubled in size.
4. Make the apple filling: Heat butter in a skillet over medium heat. Add apples, brown sugar, and cinnamon, cooking for 5-7 minutes until apples soften and the mixture becomes syrupy. Set aside to cool.
5. Assemble the cinnamon rolls: Punch down dough and roll into a 12×18-inch rectangle. Spread apple mixture evenly over the dough, leaving a border along edges. Roll tightly from the longer side, then slice into 12 equal rolls. Place in a greased 9×13-inch dish and let rise for another 30-45 minutes.
6. Preheat the oven to 350°F (175°C). Bake for 25-30 minutes, or until golden brown and cooked through.
7. Make the icing: Whisk together powdered sugar, milk, and vanilla extract until smooth.
8. Once baked, let the rolls cool for 5 minutes. Drizzle with icing and serve immediately.