Special equipment: 2 large cast-iron or other heavy skillets; 2 bricks wrapped in foil
For the salsa verde: Put the parsley, tarragon, cornichons, garlic, anchovies, vinegar, capers and some salt and pepper in a food processor. Pulse a few times to coarsely chop the herbs. With the machine running, drizzle in the olive oil to make a thick but pourable sauce. Taste and adjust the seasoning as needed.
For the pressed chicken: Heat a charcoal or gas grill to medium-low.
Season the chicken on both sides with salt and pepper. Line up the breasts skin-side down on the grill and lay a piece of foil on top to cover; do the same with the thighs. Set a heavy skillet on top of the breasts and a second skillet atop the thighs. Weight each skillet with a foil-wrapped brick. Cover the grill and cook, checking for color once or twice, until the skin is deep golden-brown and crisp, 15 to 20 minutes. Remove the bricks, skillets and foil and set aside. Flip the chicken, cover the grill and cook until the chicken reaches an internal temperature of 160 degrees F, another 5 to 10 minutes depending on thickness.
Remove the chicken to a cutting board and let rest for 5 minutes. Cut each breast through the bone into 3 or 4 pieces and transfer to a platter along with the whole thighs. Drizzle the chicken with some salsa verde. Serve the remaining salsa verde on the side.
The salsa verde can be made 1 day ahead and refrigerated, covered. Just remove it from the refrigerator 1 hour prior to serving to allow it time to come to room temperature.