Preheat oven to 375°. Rinse duck legs and pat dry. Lay legs skin side up in a roasting pan that just holds them comfortably. Sprinkle with salt, pepper, and the herbes de Provence.
Roast duck in preheated oven for 1 hour. Spoon fat from pan and save for other uses or discard. Pour wine over duck; it should be deep enough so meat is immersed but skin is exposed. Continue roasting until skin is golden red, about 30 minutes longer (20 minutes if using convection heat).
Meanwhile, pour 1 tablespoon olive oil into a 1 1/2- to 2-quart pan over medium-high heat; add shallots and stir often until beginning to brown, about 2 minutes. Add chicken broth, thyme, and bay leaf; boil, stirring occasionally, until reduced to about 1 1/2 cups, about 45 minutes. When duck is done, add 1 cup braising liquid to broth mixture and boil, stirring often, until mixture is reduced by about one-fourth, about 15 minutes. Pour through a fine strainer into a small pitcher or bowl.
While broth reduces, prepare greens: In a 12- to 14-inch frying pan over medium-high heat, melt butter with remaining 1 tablespoon olive oil. Add onion and stir often until beginning to brown, about 7 minutes. Add greens (if using mustard greens, add half, stir until wilted, then add remaining) and cook, stirring often, until tender to bite, 3 to 5 minutes. Add garlic and the anchovy, if using, and stir until fragrant, about 1 minute longer. Remove from heat and season to taste with salt, pepper, cayenne, and lemon juice.
Mound greens on plates and set duck legs on top. Serve pan juices alongside.
Wine pairing: A Pinot with both minerals and good fruit will bring out the best in the greens.