How to Make It
In a large bowl, toss 1 tablespoon of kosher salt with the cabbage and massage it until it softens and releases its liquid, about 4 minutes. Drain the cabbage in a colander and rinse it.
Wipe out the bowl. Add the vinegar, sugar, mustard seeds, 1 cup of water and 1 tablespoon of salt. Add the cabbage to the vinegar mixture, toss to coat and place a plate on top to keep it submerged. Let stand at room temperature for 2 hours.
Meanwhile, in a large saucepan, heat 1 1/2 inches of vegetable oil to 325°. In a medium bowl, toss the shallots with the Wondra flour. Fry the shallots all at once, stirring gently, until golden, about 7 minutes. Using a slotted spoon, transfer the shallots to a paper towel–lined plate to drain. Season the shallots with salt. Reserve the cooking oil.
In a medium bowl, combine the ground beef with the onion powder, garlic powder, smoked paprika, Worcestershire sauce and 2 teaspoons each of kosher salt and ground pepper. Knead gently until thoroughly mixed. Form the beef into eight 4-inch patties. Press the shredded cheese into four 2 1/2-inch disks. Sandwich the cheese disks between the patties. Pinch the edges together to seal.
Heat a grill pan or griddle. Brush the burgers with some of the shallot cooking oil and cook over moderate heat until browned on the bottom, about 3 minutes. Flip the burgers, invert a large heatproof bowl over them and cook until medium within and the cheese is melted, about 3 minutes longer.
Mound the arugula on the bun bottoms and top with the burgers. Drain the pickled cabbage and mound some of it on the burgers; reserve the rest of the pickled cabbage for another use. Top with the fried shallots and the bun tops and serve.
The pickled cabbage can be refrigerated in its brine and the fried shallots can be stored at room temperature in an airtight container for up to 4 days.
Wondra flour, which is more granular than regular all-purpose flour, is available at most supermarkets.