Submerge the pork belly in a pot just large enough to hold it, cover with cold water by 5 cm / 2 inches, and add the doenjang, coffee grounds, onion, garlic, ginger, scallion whites, bay leaves, peppercorns, and soju.
Bring to a boil uncovered, skimming off any gray foam that rises in the first 5 minutes.
Drop the heat to a bare simmer — small bubbles, not a rolling boil — cover partially, and cook 50 to 60 minutes for a 1 kg piece, until a chopstick slides through the thickest part with no resistance.
Turn off the heat and let the pork rest in the hot broth, uncovered, for 10 minutes. This keeps the outer slices from drying out while you get everything else ready.
Lift the pork out, let it firm up for 5 minutes on a cutting board, then slice against the grain into pieces about 5 mm / 1/4 inch thick.
Stir together the ssamjang ingredients in a small bowl.
Arrange the pork on a platter with the lettuce, cabbage, and kimchi alongside, and serve everyone a spoon of ssamjang for building their own wraps.