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