How to Make It
Preheat grill to very high (about 550°F). Place scallions and parsley on oiled grates; grill, uncovered, turning often, until just wilted and lightly charred, 30 seconds to 2 minutes. Remove from grill. Remove and discard parsley stems. Place scallions, parsley leaves, basil, mayonnaise, lemon juice, anchovies, garlic, 1 1/2 teaspoons salt, and 1/2 teaspoon pepper in a food processor; process until smooth, about 15 seconds. Refrigerate until chilled, about 30 minutes.
Meanwhile, brush corn evenly with 1 tablespoon oil; sprinkle with 1/4 teaspoon salt and 1/4 teaspoon pepper. Toss together shrimp, 2 tablespoons oil, remaining 1 teaspoon salt, and remaining 1/2 teaspoon pepper in a large bowl. Place corn and shrimp on oiled grates; grill, uncovered, turning occasionally, until corn is lightly charred and shrimp is just cooked through, 2 to 3 minutes for shrimp and 6 to 8 minutes for corn. Remove from grill; let cool slightly, about 5 minutes. Cut corn kernels off cobs; discard cobs. Set corn kernels and shrimp aside.
Brush cut sides of lettuces with remaining 2 tablespoons oil. Place lettuces, cut sides down, on oiled grates; grill, uncovered, until grill marks appear but lettuce is still mostly crisp and not yet wilted, 2 to 3 minutes. Remove from grill; let cool slightly. Chop into 1 1/2-inch pieces.
Place corn kernels, shrimp, lettuces, almonds, and 1 cup dressing in a large bowl; toss until evenly coated. Add avocado; season with salt and pepper to taste, and gently toss to combine. Serve with remaining dressing.