Dissolve yeast and 1/2 teaspoon sugar in warm water. Let stand for 5-10 minutes or until foamy.
Add milk, shortening, 1/3 cup sugar and salt to a saucepan. Heat over medium-high heat until very hot and sugar and salt are dissolved and shortening has melted. Let cool to 130°F.
In the bowl of an electric mixer fitted with the paddle attachment, lightly beat the eggs. With the mixer running on low, add the hot milk mixture in a slow stream.
Measure 3 cups of the gluten free flour mix into a bowl. Add the xanthan gum and whisk to combine. Add all at once to the mixer and turn on low. Turn off the mixer and add the yeast. Turn mixer back on to low speed. Add the remaining gluten free flour mix a little at a time until the dough is still a little sticky, but not stiff. (You should only need 3 3/4 to 4 cups total. Reserve the remaining for rolling out the dough.)
Beat the dough on medium speed for 4-5 minutes. Turn off the motor, remove the paddle attachment and let dough stand for a few minutes. The dough will firm up a bit as it stands, so don't skip this step.
Grease a 9x13-inch baking dish with butter. Set aside.
Lay a large rectangle of parchment paper out on a clean, flat surface. Sift a few tablespoons of the remaining flour in an even layer over the parchment paper. Turn the dough out onto the parchment paper and sift a little more flour mix over the top. Use your hands to flatten the dough into a large rectangle, about 16x20-inches and 1/2-inch thick. Trim and patch the edges as needed so they are as straight as possible.
Carefully spread the softened butter over the entire surface of the dough. Mix the cinnamon and brown sugar together and sprinkle evenly over the butter.
Start on the long edge and use the parchment paper to help you roll the dough over itself. When you get to the edge, dab a little water along the length of the dough to help seal it. Press the dough, if needed, so the roll is even through the middle.
Using a sharp serrated knife, carefully cut the roll into quarters and then cut each quarter into thirds. Transfer each roll to the buttered baking dish. Press down gently on each roll so it touches the one next to it. Brush the tops of the rolls with the 2 tablespoons melted butter. Cover baking dish tightly with plastic wrap and place somewhere warm to rise for 30-45 minutes depending on the temperature of the room. The rolls will raise up, but not double in size.
While rolls are rising, preheat oven to 425°F. Bake rolls for 20-25 minutes, or until the tops are golden brown. Remove from oven and let cool slightly before serving.
Prepare glaze: Mix all ingredients together; beat until smooth. Add water or milk to thin consistency if needed. Top rolls with vanilla glaze.