Steps:

1. 1 Cover 2 large chicken breasts with 4 cups cold water in a large saucepan.
2. 2 Season with 3 tablespoons salt.
3. 3 Bring to a gentle boil over medium heat, then immediately turn chicken with tongs. Cover pot with a tight-fitting lid, remove from heat, and let sit until an instant-read thermometer inserted into the thickest part of breast registers 150°, 5–10 minutes (depending on thickness of the breast). Transfer chicken to a cutting board and let rest for 10 minutes.
4. 4 Meanwhile, cut 1 lemon in half and squeeze cut side up into a medium bowl. Whisk in 2 tablespoons Dijon mustard. Whisking constantly, very slowly add in 6 Tbsp. olive oil until dressing is thick and emulsified. Season with salt and pepper.
5. 5 Trim ends off of 8 oz. green beans and transfer to a large resealable plastic bag. Seal bag and whack beans with a rolling pin to split the skins and soften the insides without completely pulverizing the flesh.
6. 6 Drizzle 2 tablespoons dressing into bag. Massage with your hands to break beans down further and coat. Season with salt and let sit at room temperature until ready to use.
7. 7 Slice 1 small bunch chives into 2 inch long batons. Transfer to a medium bowl.
8. 8 Separate lettuce leaves from 1 large head of Bibb lettuce and add to bowl with chives.
9. 9 Add 1 cup basil and toss to combine. Arrange on one side of a large platter. Cover lettuce and herbs with damp paper towels and chill until ready to use.
10. 10 Rinse 1 bunch radishes, trim, and slice in half (or into ½ inch wedges if large). Transfer to a medium bowl.
11. 11 Tear 1 cup whole peperoncini into bowl. Mix in 1 Tbsp. dressing; season with salt. Let sit at room temperature until ready to use.
12. 12 Slice chicken into ¼"-thick slices.
13. 13 Halve 1 avocado, remove pit, and thinly slice. Fan out slices.
14. 14 Arrange lettuce, green beans, radish salad, avocado, and chicken separately on platter. Drizzle more dressing over and serve remaining dressing alongside.