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.