Pound chicken breasts between sheets of plastic wrap to 3/4-inch thickness. Make chicken marinade: Combine 1/3 cup oil, 2 tablespoons lemon juice, herbes de Provence, and one pressed clove of garlic in large Ziploc bag. Add chicken to bag and coat with marinade. Let marinate in the refrigerator 2 to 6 hours.

Heat grill to medium-high. Remove chicken from marinade and salt and pepper chicken. Grill chicken until cooked through, about 4 minutes per side. Cool. Cut chicken on diagonal into 1/2 inch-thick slices. Whisk anchovy paste, Dijon mustard, remaining lemon juice, and remaining garlic in small bowl. Gradually whisk in remaining 3/4 cup oil. Add salt and pepper to taste. Cut both bread loaves horizontally in half. Tear out and some of the interior crumbs from all halves to create a slight well. Drizzle some dressing over inside of each top and bottom. Divide capers, chicken, onion, greens and hardboiled eggs between both sandwich bottoms. Cover with tops and press to compact. Wrap sandwiches tightly in foil, and refrigerate at least 2 hours-the sandwiches are better the longer they sit. When ready to leave for picnic, place in bottom of picnic basket or cooler so the weight of the other items presses them together. (Pan bagnat can be kept at room temperature for up to one hour.) To serve, cut into wedges.