How to Make It
In a medium saucepan, combine the white vinegar, yellow onion, garlic, whole cloves, star anise, coriander and caraway seeds and 2 1/2 tablespoons of salt and bring just to a boil, stirring to dissolve the salt. Add the cucumber slices and dill, remove from the heat and let cool completely. Transfer the cucumbers and brine to a jar and refrigerate for at least 1 hour or up to 3 days.
Meanwhile, in a blender or mini food processor, combine the egg yolks with the cider vinegar and 2 tablespoons of water and puree until smooth. With the machine on, add the oil a few drops at a time until the mayonnaise starts to thicken, then add the remaining oil in a very thin stream until the sauce is emulsified. Add the cheese and mustard powder and puree until smooth. Season the mayonnaise with salt and pepper and scrape into a bowl. Refrigerate until chilled, about 30 minutes.
Heat a cast-iron grill pan until very hot. Brush the cut sides of the buns with butter and grill over moderately high heat until lightly browned, about 1 minute; transfer to a platter. Add the red onion slices to the pan and grill until lightly charred on the bottom, about 2 minutes; transfer to a plate.
Gently form the ground beef into four 3/4-inch-thick patties, packing them as loosely as possible. Season generously with salt and pepper and grill over moderately high heat, turning once, until lightly charred on the outside and medium-rare within, about 4 minutes total. On each bun, set 3 pickle rounds, 1 slice of tomato and 1 slice of grilled onion and top with the burger and an iceberg leaf. Generously brush the bun tops with the cheddar mayonnaise, close the burgers and serve right away.