Pull skin from duck, cutting where needed to release.
Lay skin flat in a 9- or 10-inch-wide pan. Bake in a 350° oven until browned, about 30 minutes; turn skin over several times. Drain on paper towels. Pour fat from pan into a container; reserve. Wipe pan clean; set aside. Reduce oven temperature to 300°.
As skin bakes, mix salt and sugar and rub all over duck breasts. Put meat in a bowl and let stand 15 to 20 minutes, then rinse well and pat dry. Discard liquid that forms.
Rinse and drain escarole, trim out core, and coarsely chop leaves.
Lightly brush one side of each bread slice with 2 teaspoons reserved duck fat. Set bread, fat side up, in a single layer in a 10- by 15-inch pan. Bake in the 300° oven until pale gold and crisp, 20 to 25 minutes. Keep warm.
Put 2 teaspoons reserved duck fat in a 10- to 12-inch ovenproof frying pan over high heat. When hot, add breasts and brown on each side, about 6 minutes total. Put duck and pan in the 300° oven. Bake until breasts are just pink in the center (130° to 140°; cut to test), about 5 minutes. Remove from oven. Keeping breasts warm, let stand at least 10 minutes for juices to settle.
Meanwhile, finely dice duck skin, return it to its pan, and put in the oven. Bake until sizzling, about 15 minutes. Keep warm.
When breasts have rested at least 5 minutes, put 1 teaspoon duck fat and the chicken broth in a 5- to 6-quart pan over high heat. Add escarole and stir until barely wilted, about 2 minutes.
Using a slotted spoon, transfer escarole to 4 warmed dinner plates.
Slice duck breasts across the grain and lay equal portions on plates. Sprinkle with duck skin. Accompany with toast, wasabi, and pickled ginger.
Per serving without skin: 531 cal., 36% (189 cal.) from fat; 37 g protein; 21 g fat (6 g sat.); 47 g carbo (7 g fiber); 898 mg sodium; 167 mg chol.
Per serving without skin and toast: 183 cal., 11% (20 cal.) from fat; 31 g protein; 2 g fat (5 g sat.); 10 g carbo (7g fiber); 466 mg sodium; 150 mg chol.