Preheat the oven to 350 degrees Fahrenheit. Line the bottom of a 9x13 baking pan with parchment paper and set aside.
In a small dish, combine the butter and sugar for the cinnamon sugar. Add in the flour and cinnamon and set aside.
In a large mixing bowl or bowl of a stand mixer, add the melted then cooled butter, white sugar and brown sugar. Beat to combine for about 30 seconds.
Add in the eggs, greek yogurt, milk and vanilla. Beat just to combine.
Sift in the gluten free flour, baking powder, baking soda and salt. Let the batter sit for about 5 minutes. It will be on the thicker side.
Transfer to the prepared baking pan and spread evenly. Use a spoon to add dollops of the cinnamon sugar mixture on top throughout the cake. Use a knife or chopstick to swirl into the cake batter.
Bake for 35-40 minutes or until a toothpick comes out clean. Let cool for just about 10 to 15 minutes.
Meanwhile, whisk together all ingredients for the icing. Drizzle over the warm cake and let cool for a bit. Slice and serve!