Barbecued Pork Sandwich

Place garlic, onions, carrot, celery, cumin, and salt in the bowl of a food processor; process until smooth. With the motor running, slowly add 1/2 cup water; process until well combined. Place pork in a large, deep roasting pan. Rub pork generously with vegetable puree; cover and refrigerate for at least 2 hours and up to overnight.

Preheat oven to 350 degrees.

Add enough water to the roasting pan so that it comes halfway up the sides of the pork. Cover pan tightly with parchment paper-lined aluminum foil and transfer to oven. Bake until pork is very tender and falling apart, about 5 hours, checking pan occasionally to make sure water has not evaporated and adding more water as necessary.

Remove pork from oven. Let stand until cool enough to handle. (At this point, pork may be kept refrigerated and covered for up to 5 days.)

Increase oven temperature to 400 degrees.

Shred pork with a fork or pull apart using your fingers; discard any excess fat. In a large baking dish, combine shredded pork with 3 cups barbecue sauce. Transfer to oven and cook until mixture is heated through and bubbly, 5 to 10 minutes.

Divide pork and shredded lettuce evenly among rolls. Serve with remaining cup of barbecue sauce, pickles, and coleslaw.