Skillet (cast iron, non-stick, stainless steel all work)
2slicesthickly sliced bread(I use Japanese style Milk Bread but Texas style toast also works)
1-2Tbsp unsalted butter
salt + pepper
1cupcabbage, thinly sliced(preferably Korean cabbage but green cabbage also works)
1/2cupcarrot, grated or julienned
1green onion, chopped
1/4cupcabbage, thinly sliced(optional)
mayonnaise(I use Kewpie mayo)
Sriracha(optional; ketchup is a good substitute)
Heat skillet to medium heat. Melt 1 Tbsp butter and toast bread on one side. If needed, add 1 more Tbsp of butter and flip, toasting bread on other side. Set aside on a wire rack to cool slightly. (You can also "tent" the two pieces of bread so they don't steam and get soggy.)
In a large bowl, whisk egg with a fork. Add a pinch of salt and pepper and whisk a little more.
Add cabbage, carrot, and green onion. Mix with fork until all the veggies are coated with egg. The egg should completely coat the veggies, with a little egg pooling on the bottom of the bowl.
Heat skillet to medium heat. If the skillet looks dry, melt additional 1 Tbsp butter. Swirl to coat the bottom of the pan. (This prevents the egg from sticking to the pan. If there's enough butter left in the pan, don't add it.) Add eggy mixture. Using a metal spatula, shape the eggy filling into a square about the same size as the toast. I pat each side, making sure the veggies stay piled up. Cook until the eggy edges look firm and the veggies have wilted a bit, about 3-4 minutes.
Using a large metal spatula, carefully flip the eggy filling. Press with the spatula, to flatten. Cook for another 3-4 minutes, until egg is cooked through. Add 1 tsp sugar directly over the top.
Push cooked egg to one side of the skillet. On the other side, add ham and briefly heat on both sides, about 1 minute total. Add cheese and turn off the heat. The residual heat should melt the cheese, about 1-2 minutes. Transfer the ham and cheese to the top of the cooked egg. Add a generous drizzle of sriracha (or ketchup) and mayo.
Transfer to bread, adding more cabbage if desired. Top with remaining slice of bread. Cut in half and enjoy!
*This recipe easily duplicates. If cooking multiples, I recommend using a large, rectangular grill pan for faster cooking.