Make the pickled cabbage: Combine the vinegar, sugar and 1 teaspoon salt in a medium saucepan. Add the cabbage and bring to a boil. Reduce the heat; simmer until the cabbage is brightly colored and tender, about 30 minutes. Remove from the heat; stir in the jam.
Meanwhile, make the remoulade: Mix the mayonnaise, mustard, relish, capers, curry powder, turmeric and 1 teaspoon minced onion in a small bowl (reserve the remaining onion). Refrigerate until ready to use.
Make the patties: Combine the beef, half-and-half, breadcrumbs, nutmeg, remaining minced onion, 1 teaspoon salt and 1/4 teaspoon pepper in a large bowl and combine with your hands. Form the mixture into six 3/4-inch-thick patties.
Heat 1 tablespoon olive oil in a cast-iron skillet over high heat. Add 3 patties to the skillet; cook until browned, 4 minutes. Flip and cook until cooked through, about 5 more minutes. Top each burger with a slice of cheese; cover and cook until melted, 1 minute. Transfer to a plate and repeat with the remaining 1 tablespoon olive oil, burger patties and cheese. Spread the remoulade on the top buns. Sandwich with the burgers and pickled cabbage.
Photograph by Ryan Dausch
Recipe courtesy of Cynthia Wilson for Food Network Magazine