Use a sharp knife to shred the cabbages into thin slices. Choose a large bowl and toss the shredded cabbage with one teaspoon of salt in it. Start to massage it with your hand, and continue until it softens, for about 3-4 minutes. It will release its liquids, and for that, drain it in a colander and rinse it. Set it aside.

Use the same bowl, and add the white vinegar, sugar, mustard seeds, one cup of water, and one tablespoon salt. Give them a mix, and add the rinsed cabbages to the mixture. Coat the slices thoroughly with the mixture and cover the bowl by placing a plate on top. This way, you will keep it submerged. Leave it at room temperature for about 2 hours.

Meanwhile, choose a large saucepan, and heat about 2 tablespoons of frying oil over medium heat to 350 F. Use a sharp knife to slice the shallots very thinly and separate the rings. Toss the onion slices in a large bowl, add the instant flour, and thoroughly coat them. Fry the shallots all at once, stirring constantly and gently until caramelized golden for about 7 minutes. Line a plate with parchment paper, and use a slotted spoon to transfer the caramelized shallots to the plate. The excess oil will be drained. Sprinkle the shallots with salt. Reserve the oil.

Choose a medium bowl, and put the ground beef in it. Add in the onion powder, garlic powder, sweet smoked paprika, Worcestershire Sauce, one teaspoon of salt, and one teaspoon of black pepper. Knead the mixture until combined thoroughly. Then with your hands, form the beef mixture into eight 4-inch patties. Press the Cheddar Processed Cheese slices into four two 1/2-inch disks. Place the cheese and sandwich the slices between the patties. Pinch the edges together to seal. At last, you will have four burgers, ready to be cooked.

To cook the burgers, heat the grill pan or griddle. Brush the burgers with the reserved oil, and place them on the heated pan over medium heat for about three minutes until browned. Flip the burgers, use a heatproof bowl, and place it over them for a better texture. Wait for about three minutes till the cheese melts, and they are cooked through.

Place the arugula on the bottom of the toasted buns, and top with cooked burgers. Drain the pickled cabbages, and mound some of them on the burgers. You can store the leftover pickled cabbages. Top with the fried shallots and the bun tops and serve immediately. You can side these delicious burgers with Fries, Green Salad, and your favorite dressing like Kalleh Mustard Sauce, or Kalleh Garlic Sauce. Enjoy!