For the slaw: Whisk together the ketchup, mayonnaise, vinegar, celery seeds, paprika and some salt and pepper.
Put the cabbage, carrot, green onions and Spanish onions in a large bowl; season with salt and pepper, toss to combine. Pour the dressing over the vegetables and toss again. Cover and refrigerate for at least 30 minutes and up to 8 hours.
For the ketchup: Whisk together the ketchup and chipotles in a small bowl.
For the mustard: In another small bowl, whisk together the mustard and horseradish.
For the burgers: Heat a gas or charcoal grill to high. Form the ground beef into four 3/4-inch-thick patties by tossing the meat from hand to hand and flattening gently. Make a well in the center of each patty with your thumb. Brush the patties with oil and season generously with salt and pepper.
Grill the burgers until slightly charred and crusty, about 3 minutes (5 minutes for turkey, which must be cooked through to serve). Flip and continue grilling until slightly charred and crusty on the other side, 4 to 5 minutes longer for medium doneness (another 5 minutes for turkey). During the last minute of cooking, top each burger with two slices of cheese; cover the grill and cook until the cheese has melted.
Transfer the burgers to a platter. If desired, toast the buns on the grill, flat-side down, for about 15 seconds, or until golden. Place the burgers on the bun bottoms. Top with some chipotle ketchup, horseradish mustard, Russian slaw dressing and a few potato chips, if using. Set the bun tops on top and serve immediately.