Preheat the oven to 425°. Set the chicken on a rack in a roasting pan and stuff the cavity with the lemon half, garlic, thyme, 1 bay leaf and a large pinch each of salt and pepper. Rub the chicken skin all over with the olive oil. Roast the chicken for about 1 1/4 hours, or until golden brown and just cooked through.
Meanwhile, bring a large saucepan of water to a boil. In a medium saucepan, cover the potatoes with cold water. Add the remaining 2 bay leaves to the potatoes and bring just to a simmer over moderately high heat. Reduce the heat to low and simmer until the potatoes are just tender, about 20 minutes. Drain and let cool slightly. Peel the potatoes and slice them 1/4 inch thick, then transfer them to a bowl and keep warm.
Add salt to the large saucepan of boiling water, then add the haricots verts and cook, stirring often, until just tender, about 4 minutes. Drain and refresh with cold water. Pat thoroughly dry.
Discard the lemon, garlic and herbs from the chicken and tip the juices from the cavity into the roasting pan. Transfer the chicken to a carving board and let rest for 10 minutes. Discard about 1/4 cup of rendered fat from the juices. Set the pan over 2 burners over moderate heat and bring to a boil. Add the sherry vinegar, then add the sliced potatoes and the beans, tossing well to coat with the pan juices. Remove the pan from the heat and gently stir in the olives, capers and mint. Carve the chicken and add any accumulated juices to the salad. Season the salad with salt and pepper and serve at once with the chicken.
Pair with a light-bodied, fruit-forward red like Château de Pommard Burgundy from France, Henry Estate Barrel Select Umpqua Valley Pinot Noir from Oregon or Château de Pizay Morgon cru Beaujolais from France.