1To 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
2Cool 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.
3To prepare the slaw: Soak the shallots in cold water for 5 minutes
4Toss the cabbage, shallots and 1 1/2 teaspoons salt in a colander set in the sink
5Set aside to drain for at least 30 minutes and up to 1 hour.
6Rinse the cabbage and pat dry. Toss with the peanut oil and vinegar in a large bowl and season with salt.
7To cook pork: Preheat a grill to medium-high heat.
8Drain and pat the tenderloins dry, then brush with the olive oil and sprinkle with pepper.
9Grill 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.
10Set the tenderloins on a cutting board to rest for 5 minutes before slicing.
11To assemble: Toast the rolls on the grill.
12Brush the bottom half of the toasted rolls with the Creole mustard and mayonnaise.
13Lightly melt the cheese on the top halves of the toasted bread.
14Thinly slice the tenderloins and layer on the bread with the slaw, pickles, and pickled peppers, if desired. Serve.