In a stand mixer, combine the gluten free all-purpose flour, instant yeast, granulated sugar, and salt. Pour in the warm milk, melted butter, and eggs, mixing on medium speed for 3 minutes until smooth.
Cover the dough and let it rest for 20 minutes. Then, place the dough between two sheets of parchment paper and roll it into a large rectangle.
Spread the softened butter over the dough, then evenly sprinkle the brown sugar and ground cinnamon on top.
Carefully roll the dough into a log using the bottom parchment paper to help lift it. Slice into 12 rolls and place them in a greased 9x13 baking dish.
Cover and let rise in a warm spot for 45 minutes. Preheat oven to 350 degrees fahrenheit.
Bake the rolls for 25 to 30 minutes until golden brown. While they bake, whisk the powdered sugar, milk, and vanilla extract together.
Let the rolls cool slightly, then drizzle the icing over the top and serve warm.