Preheat the oven to 180°C and lightly grease an ovenproof baking dish with a little oil.
Grind the oat flakes in a blender or food processor until they resemble coarse flour, then transfer to a large bowl.
Pour the milk over the ground oats, stir well and leave to stand for 5–10 minutes so the mixture thickens slightly.
Peel and grate the carrot, then grate or finely dice the apple and set both aside.
In a separate bowl, whisk the eggs with the stevia until smooth and evenly combined.
Add the whisked eggs, baking powder, ground cinnamon and vanillin or vanilla extract to the soaked oats and mix until uniform.
Fold in the grated carrot, apple and cranberries or lingonberries, stirring until the fruit and vegetables are evenly distributed.
Transfer the mixture to the prepared baking dish and smooth the surface with a spatula or spoon.
Bake at 180°C for about 45 minutes, until the top is set, lightly golden and a skewer inserted in the centre comes out clean.
Allow the oat bake to cool for at least 15–20 minutes, then slice into portions and serve warm or chilled.