To prepare the gumbo: Prick duck skin all over with tip of a knife. In a heavy skillet brown kielbasa and duck in batches over moderately high heat, transferring as browned to paper towels to drain. Combine kielbasa and duck with broth and water in a 6-gallon kettle and bring to a simmer.
Pour off all but 1/4 cup fat from skillet. Add flour and cook roux over moderately low heat, stirring constantly, until a shade darker than peanut butter, about 30 minutes. Add onions, celery, and bell peppers and cook, stirring occasionally, until vegetables are crisp-tender. Add vegetable mixture to kettle and stir until roux is dissolved.
Simmer gumbo, uncovered, 2 hours and let cool completely. Bone duck, discarding skin and bones. Chill gumbo overnight. Gumbo may be made 2 days in advance and kept covered and chilled. Discard fat on surface and reheat gumbo with cayenne, scallions, and salt to taste over moderate heat.
To prepare the rice: To a kettle of boiling salted water add brown rice, stirring, and boil 15 minutes. Drain rice in a large colander and rinse. Put colander over a kettle of boiling water and steam brown rice, covered with a kitchen towel and a lid, until fluffy and dry, about 25 minutes. Cook white rice in same manner, boiling 10 minutes and steaming 15 minutes. In a bowl toss brown and white rice together. Rice may be made 3 days in advance and kept chilled in resealable plastic bags. Steam rice to reheat.