Preheat the oven to 375 degrees F. Spray a baking dish with nonstick spray. I used a 10×7 dish, but an 8×8, 9×9, round cake pan or another size close to that will work too.
In a large bowl, combine the oats, sugar, baking powder, salt and cinnamon. Stir well to combine. Add the egg, eggnog, coconut oil and vanilla extract, mixing well until combined. Pour the mixture in the baking dish and bake for 25 to 30 minutes, or until the oats are just set.
To caramelize the bananas, heat a large skillet over medium-high heat and add the butter. Sprinkle the bananas with the cinnamon then add them to the skillet. Sprinkle the brown sugar on top of the bananas and cook them for only 1 to 2 minutes, until they are golden but not mushy, turning once or twice. If needed, do this in two (or even three) batches.
To make the vanilla bean brown butter, add the butter to a saucepan and heat it over medium heat. Whisk constantly until the butter begins to bubble and brown bits form on the bottom. The minute you see the brown bits, remove the skillet from the heat and continue to stir for 15 seconds. Scrape the vanilla beans out of the pods and whisk them into the brown butter.
To serve the oatmeal, add the bananas on top and drizzle on a little extra eggnog. Drizzle the brown butter on top.