In a small bowl, combine the warm milk and a tablespoon of the sugar. Sprinkle the yeast on top and let it sit for about 5 to 10 minutes until it becomes foamy. Once active, whisk in the melted butter, the egg, and the alcohol-free vanilla extract until well combined.
In a large mixing bowl or the bowl of a stand mixer, whisk together the gluten-free flour, the remaining sugar, and the salt. Slowly pour the wet ingredients into the dry ingredients while mixing on low speed. Increase the speed to medium and beat the dough for about 3 to 5 minutes. Gluten-free dough will be stickier and more like a thick cake batter than traditional bread dough, which is completely normal.
Place the dough onto a large piece of parchment paper that has been lightly dusted with gluten-free flour. Place another piece of parchment on top and roll the dough into a rectangle about 1/4 inch thick. Remove the top parchment paper and spread the softened butter across the dough, leaving a small margin at the edges. Sprinkle the mixture of brown sugar and cinnamon evenly over the butter.
Using the bottom parchment paper to help you, carefully roll the dough into a tight log. Use a sharp serrated knife or unflavored dental floss to cut the log into 12 even rolls. Place the rolls into a greased 9 by 13 inch baking pan. Cover the pan with a warm, damp cloth and let the rolls rise in a warm, draft-free spot for about 45 to 60 minutes, or until they have noticeably puffed up.
Preheat your oven to 350 degrees Fahrenheit. Once the rolls have finished rising, bake them for 22 to 27 minutes. They should be lightly golden brown on the edges and firm to the touch. Avoid overbaking, as gluten-free breads can dry out quickly if left in the heat too long.
While the rolls are baking, beat together the softened cream cheese, butter, powdered sugar, and alcohol-free vanilla extract until the icing is smooth and creamy. Let the rolls cool for about 5 to 10 minutes after removing them from the oven, then spread the frosting generously over the top so it melts slightly into the layers. Serve immediately while warm.
