For the shrimp: Put 1 cup of water, chopped ginger, salt, sugar, garlic, and pickling spice in a small saucepan. Bring to a boil, then pour into a large bowl. Add 1 cup of cold water and set brine aside to cool completely. Add shrimp and refrigerate for 30 minutes.

For the salad: Strip the peel from half of 1 of the grapefruits with a vegetable peeler, taking care not to include the bitter white pith. Warm the grapefruit peel, sliced ginger, and olive oil in a small saucepan over medium heat. As soon as the oil starts to bubble, about 2 minutes, remove from heat. Set oil aside to steep 30 minutes. Strain and reserve the oil.

Prepare an outdoor grill with a medium fire.

Segment both grapefruit over a bowl, reserving the juice (see page 000). Whisk 3 tablespoons of the reserved grapefruit juice with the vinegar, soy sauce, honey, and 12 teaspoon salt in a medium bowl. Gradually whisk in the infused grapefruit oil, starting with a few drops and then adding the rest of the oil in a steady stream to make a slightly thick dressing. Season with black pepper to taste.

Remove shrimp from brine and pat dry. Brush shrimp with olive oil and grill, turning once, until shrimp are pink and just cooked through, 3 to 4 minutes.

Gently toss the shrimp, grapefruit segments, lettuce, basil, and 12 teaspoon serrano with 14 cup of the dressing. Gently toss in avocados and divide the salad among 4 plates. Sprinkle with the remaining serrano, if desired, and drizzle with a little more of the dressing.