Pulled Pork Sandwich

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1 boneless pork shoulder (about 4 to 41/2 pounds)

4 tablespoons kosher salt

1 tablespoon freshly ground black pepper

1 1/2 cups light brown sugar

1/4 cup paprika

2 to 3 sprigs thyme, leaves only

4 cloves garlic

1/4 cup red wine vinegar

Scant 1 tablespoon cayenne

3 tablespoons extra-virgin olive oil

Coleslaw:

1 tablespoon whole-grain mustard

1 cup mayonnaise

1/4 cup sour cream

1/2 lemon, juiced

2 tablespoons red wine vinegar

2 teaspoons sugar

1/2 head savoy cabbage finely sliced

1/2 head purple cabbage, finely sliced

2 green onions, chopped

2 carrots, sliced on mandoline

Kosher salt and freshly ground black pepper

6 brioche hamburger buns

1/4 bunch flat-leaf parsley, for garnish