Combine all sauce ingredients in a small bowl or squeeze bottle, mix thoroughly and set aside
Prepare vegetables. Thinly shred cabbage, thinly shred or slice carrot, thinly slice green onion. Add all vegetables to a bowl and mix. Take two handfuls of slaw mixture out of bowl and set aside.
Add eggs to bowl with vegetable mixture, stir to combine. Season with salt and pepper.
Heat a large pan over medium heat. Use stick of butter directly on pan (like a glue stick) and coat pan with butter. Add bread to pan and toast bread until golden brown about 2 minutes then flip and cook for about 2 more minutes, applying more butter to pan if needed. Remove bread from pan and set aside.
Spread more butter on the pan. Add egg mixture to one half of the pan, pushing it towards the edge to form a long rectangle shape.
Cook for about 3 minutes until bottom is firm. Using a wooden spatula, split the egg in the middle to form two smaller squares. Flip the egg and place one slice of cheese on top of each egg. Cook for 3 more minutes.
Add ham to other half of pan and heat through for about 1 minute, flipping once. Remove ham from pan. At this point, the egg should be done. If cheese is taking too long to melt, you can cover the pan with a lid until cheese melts, about 1 minute.
Get your toasted bread ready, top with egg, ham, reserved cabbage mixture. Drizzle with plenty of sauce. Serve!