Combine 1 cup of flour, granulated sugar, active dry yeast, and salt in a large mixing bowl. Heat whole milk and melted butter until warm, then pour into dry ingredients. Add the egg and mix until combined. Gradually add remaining flour until a soft dough forms.

Prepare the cinnamon filling by mixing brown sugar and ground cinnamon in a small bowl. Set aside.

After the dough has risen for about an hour, punch it down and roll it out into a rectangle. Spread softened butter over the dough, sprinkle with cinnamon filling, roll tightly, and cut into 12 pieces. Place in a greased baking dish and let rise for another 30 minutes.

Preheat the oven to 350°F. Bake the rolls for 25-30 minutes until golden brown. Let cool for 10 minutes.

Prepare the icing by whisking together powdered sugar, milk, and vanilla extract until smooth. Drizzle over warm rolls before serving.