Step 1Remove 4 to 5 layers of the large outer leaves of the lettuce, until the remaining head is compact and light green. Cut the heads in half lengthwise.
Step 2Heat a grill or grill pan and brush it with 1 tablespoon olive oil. Place the lettuce cut-side down on the grill, pressing down slightly to flatten. Cook on each side until there are distinct grill marks, about 2 minutes. Remove from the pan and reserve.
Step 3Rinse the scallops and pat dry. Toss them in 1 tablespoon olive oil, one-fourth teaspoon salt and a pinch of white pepper. Set aside.
Step 4In a separate small saute pan, cook the minced shallot in one-half tablespoon butter until softened, about 1 minute. Add the wine and vinegar; cook until the liquid is reduced by half, about 1 minute. Add the chicken stock to pan and bring to a simmer. Simmer for about 5 minutes.
Step 5Remove the cores from the grilled lettuce halves and cut the halves in half lengthwise, then crosswise into strips about 1 1/2 inches wide. Add the lettuce pieces to the simmering liquid. Cook until tender, about 5 to 7 minutes.
Step 6While the lettuce is cooking, heat the grill to medium-high and brush it with the remaining olive oil. Grill the scallops about 2 minutes on each side, or until they are opaque and have distinct grill marks. Remove them from the heat.
Step 7Remove the lettuce from the cooking liquid and place it on serving plate or platter. Add a pinch of sugar to the cooking liquid and swirl in one-half tablespoon butter. Taste and add salt, pepper, sugar and/or vinegar to taste.
Step 8Top the lettuce with grilled scallops and pour the sauce over. Serve warm.