1 gallon water
3 cups coarse kosher salt
1 6 1/2-pound bone-in pork shoulder roast (Boston butt), excess fat trimmed
3 large heads of garlic, halved crosswise
1 1/2 teaspoons whole black peppercorns
1 large head of green cabbage, cut into 6 wedges
1 pound carrots, peeled, cut crosswise in half, then quartered lengthwise
Fresh thyme sprigs
Combine 1 gallon of water and salt in heavy large pot. Stir until salt dissolves. Add pork. Cover and refrigerate 1 day.
Bring pork in salt water to boil. Boil 10 minutes. Carefully drain salt water. Fill pot with enough cold water to cover pork. Bring water to boil over high heat. Add garlic and peppercorns. Reduce heat to medium-low. Cover; simmer gently until pork is very tender, about 3 hours. Transfer pork to large pan. Cover; keep warm.
Add cabbage and carrots to cooking liquid. Boil until vegetables are almost tender, about 15 minutes. (Can be made 1 day ahead. Cool slightly. Chill uncovered until cold, then cover and keep chilled. Before continuing, rewarm over medium-low heat.) Using slotted spoon, transfer vegetables and garlic to serving platter.
Cut pork into thick slices; arrange on platter with vegetables. Spoon some hot cooking liquid over pork and vegetables. Garnish with thyme. Serve with mustard.