To brine the pork: Combine 1 1/2 cups cold water, 3 tablespoons salt, sugar, garlic, and ginger in a medium saucepan. Bring to a boil, remove from the heat, and stir in the rum. Cool to room temperature. Put the tenderloins in a bowl or shallow container and pour the brine over them. (Or put the tenderloins and brine in a large re-sealable plastic bag.) Cover and refrigerate at least 1 hour or up to 4 hours.
Meanwhile make the slaw: Soak the shallots in cold water for 5 minutes. Toss the cabbage, shallots and 1 1/2 teaspoons salt in a colander set in the sink. Set aside to drain for at least 30 minutes and up to 1 hour. Rinse the cabbage and pat dry. Toss with the peanut oil and vinegar in a large bowl and season with salt.
Preheat a grill to medium-high heat. Drain and pat the tenderloins dry, then brush with the olive oil and sprinkle with pepper. Grill the tenderloins, turning as needed to mark all sides, and until an instant-read thermometer inserted in the thickest part of the tenderloin registers 145 degrees F, about 8 minutes per side. Set the tenderloins on a cutting board to rest for 5 minutes before slicing.
While the tenderloins cook, toast the rolls on the grill. Brush the bottom half of the toasted rolls with the Creole mustard and mayonnaise. Lightly melt the cheese on the top halves of the toasted bread.
Thinly slice the tenderloins and layer on the bread with the slaw, pickles, and pickled peppers, if desired. Serve.
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