Cut the mirlitons in half lengthwise. Remove the seeds. Cut each half lengthwise into slices about 1/2-inch thick. Put the mirlitons, carrots, onions, and garlic in a large bowl. Cover with ice and cold water. Let sit for 2 hours to perk up the vegetables. Drain thoroughly. In a large nonreactive stockpot, combine the vinegar, bay leaves, salt, sugar, peppercorns, cayenne, and mustard. Bring to a boil. Add the vegetables, reduce the heat, and simmer for 5 minutes. Remove from the heat. Sterilize 7 quart-size preserving jars and keep hot. Soak the lids and keep hot. Pack each jar tightly with the vegetables, dividing them evenly, and enough of the hot liquid to come within 1/2-inch of the top. With a clean damp towel, wipe the rim and fit with a hot lid. Tightly screw on the metal ring and process in a hot-water bath for 15 minutes. Using tongs, remove the jars, place on a towel, and let cool. Test seals. Tighten the rings. Store in a cool dry place. Let age for 4 to 5 weeks.
Once pickled, mirlitons can be used in salads, as a relish or to garnish a martini.