Heat the oven to 325°F. Place onion in a small Dutch oven, add pork, salt, pepper and braising water, then cover tightly.
Braise for 3 to 3 1/2 hours until the pork is fully fork-tender and pulls apart easily, well above the 145°F minimum. Rest covered for 10 minutes, then remove excess fat and pull meat into coarse shreds, discarding tough connective pieces.
Whisk mustard, vinegar, brown sugar and Worcestershire in a saucepan and simmer for 3 minutes. Toss drained warm pork with the sauce without adding the watery braising liquid.
Divide half the pepper Jack over 4 bread slices, add the sauced pork evenly and top with remaining cheese. Close with the other bread slices and spread mayonnaise over the outside faces.
Cook 2 sandwiches at a time in a large skillet over medium-low heat for 4 to 5 minutes per side, covering for 2 minutes after turning, until the bread is golden and filling reaches 165°F. Repeat for the second batch.
Rest each batch for 2 minutes before cutting and serve one sandwich per person.