Cheddar-Stuffed Burgers with Pickled Slaw and Fried Shallots

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.