Heat the oven to 350F and oil a small baking dish, about 8 inches square.
Grate the carrots on the coarse side of a box grater. Grated fine, they melt into the bake; grated coarse, they keep a little bite and read as carrot cake.
Whisk the eggs, yogurt, cinnamon and salt together until smooth, then stir in the baking powder.
Fold in the oats and grated carrot until everything is evenly wet. Let the bowl sit 5 minutes so the oats start to drink the liquid, which is what sets the bake instead of leaving it loose.
Scrape into the dish, level the top, and bake 30 to 35 minutes until the center is set and springs back when pressed.
Cool 10 minutes before cutting. It firms as it cools, so slicing warm gives ragged edges.