Step 1

Heat a medium skillet over medium-high. Combine ginger and 1 tablespoon sugar; rub evenly over all sides of pork. Add 1 tablespoon oil to pan; swirl to coat. Add pork; cook, turning until charred on all sides, about 8 minutes. Place pork in a 5- to 6-quart slow cooker; add stock and garlic. Cover and cook on low until pork shreds easily with a fork, 7 1/2 to 8 hours. Remove pork from slow cooker, reserving 1/4 cup cooking liquid. Shred meat with 2 forks, and place in a large bowl. Add reserved 1/4 cup cooking liquid and 3/4 teaspoon salt; toss well to combine.

Step 2

Meanwhile, combine vinegar, 1 1/4 cups water, remaining 2 tablespoons sugar, and remaining 3/4 teaspoon salt in a saucepan; bring to a boil over medium-high, stirring to dissolve sugar. Add carrots and radish. Remove from heat; cool completely, about 30 minutes or up to 7 hours. Drain well.

Step 3

Combine miso, Sriracha, and remaining 1 1/2 tablespoons oil. Spread about 1 teaspoon sauce over each lettuce leaf; top each with about 1 ounce pork and 2 tablespoons pickled vegetables. Serve with rice.