Step 1

Cook green beans in a large pot of boiling heavily salted water 30 seconds. Add corn and cook until beans are bright green and crisp-tender and corn is just barely tender, about 15 seconds. Using a spider or slotted spoon, transfer vegetables to a medium bowl of ice water; keep pot of water boiling. Let vegetables cool, then drain and transfer to a large bowl.

Step 2

Add chicken to pot; return water to a gentle boil. Reduce heat and cook at a bare simmer until juices run clear when thickest part of chicken is pierced with the tip of a small knife (an instant-read thermometer inserted into thickest part of chicken should register 150°; internal temperature will climb to 160° as chicken rests), 20–25 minutes for skin-on, bone-in breasts; 10–15 minutes for skinless, boneless breasts. Transfer chicken to a plate; let cool slightly. Shred meat into big pieces; discard skin and bones if present. Add shredded meat to bowl with vegetables.

Step 3

Purée anchovy, avocado, oil, sour cream, lemon zest, lemon juice, vinegar, ½ cup herbs, and ¼ cup water in a blender, adding more water to thin if needed, until smooth and thick (dressing should be the consistency of heavy cream); season with salt and pepper. Taste and add more lemon juice, salt, and/or pepper if needed (dressing should be bright and acidic).

Step 4

Toss chicken and vegetables with half of dressing, half of fried shallots, and ¼ cup herbs. Arrange lettuce on a platter; mound chicken salad on top. Drizzle with more dressing. Top with more herbs and remaining fried shallots. Serve with remaining dressing alongside.Do ahead: Chicken can be cooked and shredded 3 days ahead. Transfer to an airtight container; cover and chill.