Trim the ends from the haricots verts. Peel the carrots, cut into three-inch pieces, and cut each piece lengthwise into slivers of the same width as the beans. Blanch the vegetables until just crisp and refresh under cold running water.

Wash and dry the chicken breasts. Flatten them slightly by pounding between sheets of waxed paper and cut into manageable pieces. Season with herbes de Provence and salt and pepper to taste on both sides. Stir the paprika into the flour in a flat bowl Dredge the chicken pieces in the flour mixture and shake off the excess.

Melt half the butter and oil in a large pan and brown the chicken on both sides over medium to high heat, adding more butter and oil as necesary. Remove the chicken to a platter in a warm oven. Meanwhile, mince the shallots. Once the chicken has browned, add the shallots to the empty pan, sauté briefly, and deglaze with half the calvados. Add the cream, bring to a boil, and return the chicken and any accumulated juices to the pan. Simmer over medium heat, turning occasionally, for five to ten minutes until the sauce is thickened. Add the vegetables and remaining calvados and stir until heated through. Remove the chicken once again to the warm platter, cover with the vegetables, and pour on the sauce.