How to Make It
Make the sauce In a small bowl, whisk together all of the ingredients until smooth.
Make the slaw In a medium bowl, whisk the vinegar with the garlic, celery, sesame seeds, sugar, fish sauce and mustard. Season with salt and pepper. Just before serving, toss with the lettuce.
Prepare the green tomato Spread the flour and panko in separate shallow bowls and season each with salt and pepper. In another shallow bowl, whisk the buttermilk and egg and season with salt and pepper. In a large nonstick skillet, heat the oil. Dredge the tomato slices in the flour, shaking off the excess. Dip in the buttermilk-egg mixture, then dredge in the panko. Fry the tomato slices over moderate heat, turning once, until golden and crisp, about 3 minutes. Transfer to a paper towel–lined plate to drain, then season with salt.
Make the burgers Form the beef into eight 1/8-inch-thick patties and season with salt and pepper. In a large cast-iron skillet set on the grate of a preheated grill or on the stovetop, heat the oil. Working in 2 batches, cook the burgers over moderate heat until browned on the bottom, about 2 minutes. Flip, top each with a slice of cheese and cook, covered, until the burgers are browned and the cheese is melted, about 1 minute longer. Transfer to a work surface and repeat with the remaining 4 patties and cheese slices.
Spread the sauce on the cut sides of the buns. Place a burger on each bottom bun. Top with a fried green tomato slice, another burger and the slaw. Close the burgers and serve.