Combine the mustard, mayonnaise and horseradish in a small bowl. Stir to combine and set aside.
Combine the spinach, cabbage and carrot in a medium bowl. Drizzle with the olive oil and vinegar. Season with salt and pepper. Toss to combine.
Lay the bread slices out on your work surface and spread with the mustard mixture. Top 4 of the slices with half of the cheese. Pile the vegetable mixture on top of each, making an even layer. Top with the remaining cheese, then the remaining bread, sauce-side down. Press in the sides of the sandwiches, if needed, to neaten them up.
Heat a large 2-burner flattop griddle over medium-low heat. Melt the butter and brush all over the griddle. Lay the sandwiches on the griddle and toast until crisp on the underside, 3 to 4 minutes. Flip and toast the second side, until the cheese is melted and oozy, about 4 minutes more.
Remove the sandwiches to a cutting board and slice in half with a serrated knife.