Grease an 8-inch round cake pan with butter or non-stick spray. If you're combining the ingredients in a separate bowl (see notes), then it doesn't hurt to put a round of parchment paper down as well.
Either in your cake pan or (as I did it, in a large mixing bowl), add the dry ingredients including the yeast and whisk to combine.
Add the milk, melted butter, oil, and beaten egg. Whisk to combine, making sure not to leave pockets of flour (though don't overmix).
Transfer the batter to your cake pan if it is in a bowl, and allow to batter to rest in the pan for 10 to 15 minutes.
While the cake rests, make the cinnamon topping. In a small bowl stir together brown sugar, cinnamon, melted butter, and salt until the mixture is moistened.
Once it's done resting, top the batter evenly with the cinnamon mixture, then use a butter knife to swirl the cinnamon sugar around through the batter. Allow to rest for 5 more minutes.
To bake the cake, place the pan in a COLD oven, then set the temperature to 350°F and bake for 25 to 30 minutes. It should be lightly browned and firm in center, and a tester should come out clean (minus some gooey cinnamon-sugar).
Allow the cake to cool for 20 minutes. Then combine icing ingredients in the same bowl used to make the cinnamon mixture and drizzle over the just warm cake.
Sprinkle generously with the purple, green, and gold (to make it king cake-like) and enjoy!