First make the sautéed cabbage. Thinly slice the onion and cabbage.
Heat the oil in a frying pan, then sauté the onion for 5 minutes. Add the caraway seeds, stir-fry for a few seconds, then add the cabbage and apple cider vinegar. Put the lid on and sauté for 10-15 minutes over low heat, stirring occasionally. Add a drop of water if it’s too dry. When the cabbage has softened, add some honey to your liking and season with salt and pepper. Set aside.
Heat a drop of oil in an other, flat frying pan (I like using pancake pans for omelettes).
Beat 3 eggs with 3 pinches of salt in a small bowl. When the frying pan is hot, pour in the mixture, letting it spread out evenly. Cook over low heat for about 5 minutes.
When the surface is no longer runny, lay 3 slices of ham on one half of the omelette, then spoon some of the cabbage on top.
If you’re using cheese, scatter some on the other half of the omelette. Cook for a few seconds, then fold the cheesy or empty half over the stuffed half with a spatula, then press down slightly. Cook for a minute or so, then slide it onto a plate.
Repeat with the leftover eggs, ham and cabbage.
Serve warm with crusty bread on the side.