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1
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.
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2
Drain the cabbage in a colander and rinse it.
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3
Wipe out the bowl.
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4
Add the vinegar, sugar, mustard seeds, 1 cup of water and 1 tablespoon of salt.
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5
Add the cabbage to the vinegar mixture, toss to coat and place a plate on top to keep it submerged.
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6
Let stand at room temperature for 2 hours.
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7
Meanwhile, in a large saucepan, heat 1 1/2 inches of vegetable oil to 325.
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8
In a medium bowl, toss the shallots with the Wondra flour.
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9
Fry the shallots all at once, stirring gently, until golden, about 7 minutes.
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10
Using a slotted spoon, transfer the shallots to a paper towellined plate to drain.
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11
Season the shallots with salt.
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12
Reserve the cooking oil.
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13
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.
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14
Knead gently until thoroughly mixed.
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15
Form the beef into eight 4-inch patties.
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16
Press the shredded cheese into four 2 1/2-inch disks.
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17
Sandwich the cheese disks between the patties.
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18
Pinch the edges together to seal.
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19
Heat a grill pan or griddle.
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20
Brush the burgers with some of the shallot cooking oil and cook over moderate heat until browned on the bottom, about 3 minutes.
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21
Flip the burgers, invert a large heatproof bowl over them and cook until medium within and the cheese is melted, about 3 minutes longer.
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22
Mound the arugula on the bun bottoms and top with the burgers.
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23
Drain the pickled cabbage and mound some of it on the burgers; reserve the rest of the pickled cabbage for another use.
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24
Top with the fried shallots and the bun tops and serve.
