Preheat the oven to 350ºF. Grease a 8X8-inch baking dish with coconut oil and line the bottom with parchment. Set aside.
In a large bowl, whisk together the oat flour, brown sugar, baking powder, baking soda, and salt. In a medium bowl, whisk together the wet ingredients. Add the wet mix to the dry mix and stir well to combine (batter will be thin).
Pour the batter into the prepared pan and let it rest for 10 minutes, while the oven heats.
Place the cake in the oven and bake for 35-40 minutes, until a tester inserted in the middle comes out clean. Remove from the oven, cool in the pan on a wire rack for 15 minutes, then turn out on the rack to cool completely.
Make the Frosting
When the cake is fully cooled, make the frosting: whisk the Greek yogurt and the lemon zest together, then sift in the powdered sugar and whisk to combine. Spread the frosting over the cake. Place the cake in the fridge if you wish to set the icing a bit.
Notes
Rest the batter - Don't skip the step of resting the batter for 10 minutes! This is what helps the cake have such a plushy, fluffy crumb because the liquid helps to hydrate the oat flour, which then results in a better structure after it's baked.
Make it dairy-free - In the cake batter, substitute non-dairy milk or non-dairy yogurt for the yogurt. In the frosting, use non-dairy yogurt for the Greek yogurt.
Make it refined sugar-free - Easily make this cake refined sugar-free by swapping coconut sugar for the brown sugar in the cake, and experiment with using maple syrup or honey in the frosting. Note that you'll use less of a liquid sweetener than the powdered sugar.
Make it egg-free - I have not experimented with making oat flour cake without eggs, but you could try substituting two flax eggs for the eggs, or even applesauce.
Bake as cupcakes - If you wish, you can make this recipe as cupcakes! Line a cupcake tin with liners, fill them ⅔ full, and bake the cupcakes for about 20-25 minutes, until a tester inserted in the middle of a cupcake comes out clean.
Try different frostings - This cake is a lovely jumping-off place for all sorts of flavor combinations. You could top it with a chocolate glaze, cream cheese frosting, a drizzle of caramel sauce, fluffy chocolate buttercream, or a simple dusting of powdered sugar.
Add different citrus - Swap the lemon zest in the icing for lime, orange, grapefruit, or mandarin zest for a flavor twist.
Make it a layer cake - Double the recipe and bake it in two parchment-lined 9-inch baking pans. Fill and frost the cake with jam, frosting, or whatever else you can think of.
Add fruit - This would make a lovely fruit cake! Stir in 1 cup of raspberries, blackberries, blueberries, or a combination into the batter.