For the quick pickles: Combine the sugar, apple cider vinegar, rice wine vinegar, dried dill, mustard seeds, red pepper flakes, garlic, onion and 1 1/2 teaspoons salt in a small saucepan. Bring to a boil.
Toss the cucumber slices with 1 tablespoon of salt and place in a strainer. Let the slices sit for 10 minutes, then rinse well with cold water.
Put the pickles in a jar, pour in the pickling liquid and leave for at least 30 minutes. Keep refrigerated for up to 10 days.
For the cheater mojo dressing: Mix together the dressing, lime juice and orange juice in a bowl. Season with salt and pepper.
For the sandwiches: Heat a large cast-iron pan over medium heat.
Spread a thin layer of mustard on the inside of each roll. Put some quick pickle slices on each bottom roll, then top with a cheese slice. (You may need to cut the cheese slices into triangles to fit.) Toss the ham in the bowl with the mojo dressing, letting the excess drip off. Put 4 ounces of ham on each sandwich. Put the tops on the sandwiches.
Spread butter on the outside of the sandwiches and put them in the cast-iron pan (do not crowd the pan; you may need to do 2 batches). Place a second cast-iron pan on top of the sandwiches and press down. Cook the sandwiches on one side for 3 to 4 minutes. Remove the top pan and turn the sandwiches over, making sure they are golden brown on the bottom. Place the top pan back on the sandwiches and press down. Cook the sandwiches until golden brown on the second side, another 3 to 4 minutes. Transfer to a cutting board and cut in half diagonally. Serve warm.
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