Slice the truffle very thinly using a mandoline or a very sharp, thin-bladed knife and a steady hand. Put the slices in a small bowl, add the Madeira, and let soak briefly to moisten the truffles (making it easier to slide them under the chicken skin and giving them some Madeira flavor). Strain the truffle in a fine-mesh strainer set over a small bowl. Reserve the liquid.
Moisten your hands under cold running water and loosen the skin on the chickens’ breasts and thighs, carefully sliding your hands between the skin and meat. Slide the truffle slices under the skin of the chicken breasts to cover them, then slide a slice or two under the skin of each thigh. Finely chop the remaining truffle slices and set aside for use in the sauce. Truss the chickens, cover with plastic wrap, and refrigerate overnight to allow the truffles to flavor and perfume the chickens.
Heat the oven to 400°F. Season the chickens generously with salt and pepper, then rub with the softened butter. Set on a rack in a roasting pan and roast for 40 minutes. (The chicken will be underdone.)
Pour the stock, truffle juice, reserved chopped truffle, and reserved Madeira into an ovenproof, flameproof casserole large enough to hold the chickens with some room to spare. Bring to a boil, then lower the heat so the liquid is simmering and cook until reduced to about 4 cups, about 30 minutes.
Remove the strings from the chickens and set them in the casserole, breast side up. Cover and braise at a simmer until fully cooked, about 20 minutes. Meanwhile, cut the carrots and leeks into square segments, keeping them separate. Wash the leeks well. Cut off the ends of the zucchini and cut out the center portion by slicing down its length to remove 4 rounded sides. Discard the rectangular center portion and cut the outer portions into 21/2-x1/4-inch-square segments.
Transfer the chickens to a large serving platter and cover with aluminum foil to keep warm. Add the carrots, leeks, zucchini, and haricots verts to the braising liquid, raise the heat to high, and cook until the vegetables are heated through and the broth has reduced a bit, about 5 minutes. Swirl in the cold butter, one cube at a time, to thicken and enrich it. Taste and adjust the seasoning with salt, pepper, Madeira, and/or lemon juice.
Spoon some sauce and vegetables over the chickens on the serving platter and serve, or return the chickens to the casserole and serve from the casserole, carving the chickens at the table and serving the pieces in bowls, spooning vegetables and sauce over each serving.