For the pickles, whisk together vinegar, sugar and salt until dissolved, about 1 minute. Add onions and let stand until ready to serve or store in the fridge for up to 2 weeks.
Bring a pot of water to a boil, add green beans and salt, boil 1 minute, then remove beans with a skimmer or spider to an ice bath. Drain and chop beans into thirds. Add whole potatoes to boiling water and cook to just fork-tender but still a bit firm, 8-10 minutes. Drain the potatoes and let cool to handle; thinly slice 1/4-inch thick.
Place eggs in a small pot, cover with water and bring to a full, rolling boil. Cover, turn off heat and let stand 12 minutes. Crack and soak eggs in cold water then peel, quarter lengthwise and slice.
Halve chicken breasts horizontally and pound each piece to 1/8-inch thick. Season with salt and pepper then sauté cutlets in EVOO, 2 turns of the pan, in a large skillet over medium-high heat for about 6-7 minutes, until browned and cooked through, turning occasionally. Remove to a cutting board and slice cutlets into thin strips across the breast on a bias.
On four large plates, arrange a bed of lettuce and top with piles of sliced chicken, tomatoes, olives, potatoes, beans, hard-boiled eggs, pickled onions and cornichons.
Whisk up dressing ingredients and stream in EVOO. Spoon dressing evenly over salads and serve with a warm baguette and good butter alongside.