Peel the banana and mash it with a fork until you get a smooth puree.
Finely chop the carrot in a blender, then transfer it to a bowl together with the banana puree and the oats. If the oats are very large, briefly pulse them in the blender as well. Mix everything together.
Add the sugar, baking powder and vanilla to the oat mixture and stir well to combine.
Wash and dry the eggs, then add them to the oat mixture. Mix again until the eggs are fully incorporated.
Pour the warm milk over all the ingredients and let the mixture rest for 15 minutes so the oats can swell and the batter thickens.
Add the ground walnuts to the batter, keeping 1–2 tbsp aside for decoration if desired.
Stir the batter well and let it rest for another 5 minutes.
Pour the oat batter into a silicone baking pan and bake at 350°F (180°C) for 20–25 minutes.
Let the carrot cake cool completely in the pan.
Flip the silicone pan over to release the cake onto a serving plate.
Dust the top with powdered sugar if desired.
Spread a thick layer of sour cream over the top and sprinkle with the reserved walnuts.
Let the dessert cool slightly, then slice it into pieces and serve.