Heat oven to 350 degrees. Grease and flour (use whatever gluten free flour you have on hand) either a 9x13 baking pan or a bundt pan. The bundt pan will look fancier, but either will work just fine.
Beat eggs, oil, milk, vanilla and butter flavorings until well mixed. Add the cake mix and stir until all ingredients are well combined.
In a separate bowl, combine the brown sugar and cinnamon.
Pour half of cake batter into prepared baking pan. Evenly sprinkle the cinnamon sugar mixture over batter. Swirl the cinnamon mixture into the batter. Pour on the rest of the cake batter. Using a knife, swirl the cinnamon around, but don't completely mix it in. Just do a good swirl. It should look about like this picture when you're finished swirling. If too much sugar/cinnamon is on top of the batter (not swirled in), it will burn during baking and just flake off. You don't want to lose any of that goodness!
Bake approximately 40 minutes for a 9x13 pan and up to 1 hour and 10-15 minutes for a bundt pan. Depending on your altitude, choice of baking pan and oven, baking times will vary. Bake until a toothpick inserted comes out clean. Allow cake to cool completely before cutting. If you used a bundt pan, allow the cake to cool 15-20 minutes before removing it from the pan. (With these gluten free cake mixes, I always err a little on the side of baking longer, not shorter. If you think you might see a tiny bit of moisture on that toothpick, bake a few more minutes.)
If you want a pretty glaze on the cake, mix about one cup of powdered sugar with a couple teaspoons of water or milk until you achieve the consistency you prefer. In my opinion, this cake is sweet enough that it doesn't need the glaze, but it does look pretty.
