Combine the warm milk, sugar and yeast in small bowl and let sit until bubbly, about 5 minutes.
To make the psyllium gel, combine the psyllium husk and water in a small bowl and let sit about 2 minutes.
Using a stand mixer with the paddle attachment, mix the gluten-free flour, sugar, salt and baking powder on low speed until combined.
In a separate bowl, whisk together the greek yogurt, egg, melted butter and water.
Slowly pour the yogurt mixture, psyllium gel and yeast mixture into the flour mixture, and mix until combined, scraping down the sides of the bowl as needed. Increase the speed to medium and beat until the dough is sticky and uniform, about 5 minutes.
Using a small cookie scoop or two spoons, scoop out tablespoon sized balls onto a lightly gluten-free floured surface. You should get around 30 balls.
To make the cinnamon sugar coating, melt ½ cup of the butter and pour into a shallow bowl or dish. In another, shallow bowl or dish, mix together the granulated sugar and cinnamon. Grease a 10-inch springform pan. With lightly floured hands, roll a dough ball into a smooth ball, coat in melted butter and then coat in the cinnamon sugar and place in the pan. Repeat with remaining dough balls. Loosely cover the pan with plastic wrap and let rise in a warm spot for 30-45 minutes. Meanwhile, preheat the oven to 350°F (180°C).
Remove the plastic wrap from over the pan. In a small bowl, whisk together the remaining ¼ cup of melted butter, brown sugar and vanilla and pour evenly over the bread.
To prevent the cinnamon sugar syrup from spilling into your oven, wrap the bottom of the pan in foil and place on a baking sheet. Bake for 35-40 minutes, until the bread is golden. Let the bread cool for about 15 to 20 minutes, still slightly warm, but not hot enough to melt the icing. Whisk together the powdered sugar, milk and vanilla and drizzle over the bread and serve.