Place olive oil and butter in a small saucepan over moderate heat until the mixture gets foamy, about 3 minutes.
Rinse the chicken, inside and out, under cool, running water and pat dry with paper towels. Fill the cavity of each bird with half the shallots, garlic cloves, thyme, tarragon, and peppercorns. Rub each chicken with half a cut lemon and put it in the cavity, too.
Using a basting brush, coat the birds, on all sides and in crevices, with butter and olive oil mixture. Place in oven and roast until skin is crisp and meat is thoroughly cooked but still juice, 55-60 minutes, or until instant read thermometer inserted in thickest part of thigh indicates 165-170 degrees. About half way through, rotate the pans so the chickens brown on all sides.
Transfer chickens to a cutting board when done, and let them rest 10 minutes before carving. To carve: use a sharp heavy knife to separate the legs and thighs and the wing portions. Then slide a knife with a thinner blade down each side of the breast bone, scooping out the breast section and leaving the skin intact. Each chicken will serve two people:overlap one leg, thigh, and wing to form a base for one side of breast meat.
Top with sauce and serve.
To make Grainy Mustard Sauce;.
Add vermouth to the chicken broth and, in a large saucepan over moderately high heat, reduce the mixture by half.
Place the heavy cream in a small bowl and whisk in about 1/4 cup of the hot broth. Slowly whisk the cream mixture back into the hot stock. Whisk in the mustard and reduce heat, simmering until slightly thickened, 3-5 minutes. Add the herbs, season with salt and pepper, and ladle over the chicken.