In a large skillet, heat the oil over medium. Add the cabbage; cook until wilted, about 2 minutes. Season with the celery seed, salt and white pepper. Douse with the vinegar. Lower the heat to keep the cabbage mixture warm.
Heat a griddle or large skillet over medium-low to medium.
In a medium bowl, combine the sour cream, ketchup, relish and scallions; season with salt and black pepper.
In a small bowl, melt the butter in the microwave or melt it in a small pan over low heat. Brush one side of the bread slices with melted butter.
Build the sandwiches with the buttered sides of the bread facing out. Spread 4 slices of bread with the mustard, top with half the cheese, the ham, the remaining cheese and the cabbage mixture. Slather the remaining 4 slices of bread liberally with green onion dressing; set on top of sandwiches, buttered side facing out.
Griddle the sandwiches, pressing down with a spatula, until the bread is golden, the cheese melts and the sandwich is hot all the way through, about 5 minutes per side.