In a large bowl, whisk together the gluten-free flour, sugar, instant yeast, and salt. Add the warm milk, melted butter, egg, and alcohol-free vanilla. Use a stand mixer or a sturdy spoon to mix until a soft, slightly sticky dough forms. Gluten-free dough is different from traditional dough, as it will be more like a thick cake batter or paste rather than a stretchy ball.
Cover the bowl with a warm, damp cloth or plastic wrap. Place it in a warm, draft-free spot and let the dough rise for about 30 to 40 minutes. It will not double in size like wheat dough, but it should look puffed and feel lighter to the touch.
Place the dough onto a large sheet of parchment paper lightly dusted with gluten-free flour. Top with another sheet of parchment and roll the dough into a rectangle about 1/4 inch thick. Remove the top sheet of paper and spread the very soft butter over 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 log starting from the long edge. Once rolled, use a piece of unflavored dental floss or a very sharp knife to slice the log into 12 even pieces. Place the rolls into a greased 9×13 inch baking pan.
Cover the pan and let the rolls rise for another 20 minutes while you preheat your oven to 350 degrees Fahrenheit. Bake for 22 to 25 minutes, or until the edges are lightly golden brown. Be careful not to overbake, as gluten-free bread can become dry quickly.
While the rolls bake, beat together the cream cheese, softened butter, powdered sugar, alcohol-free vanilla, and enough milk to reach your desired consistency. Spread the frosting over the Gluten-Free Cinnamon Rolls while they are still slightly warm so it melts into the center of each roll.
