Chef Prudhomme's approach to making a roux is quite different from most recipes. It does work if you are very careful not to burn the mixture.
Remove excess fat from chicken pieces. Rub a generous amount of salt and cayenne on both sides of the pieces, making sure each is evenly covered. Let stand at room temperature for 30 minutes. Combine the flour, 1/2 tsp salt and 1/2 tsp cayenne in a paper or plastic bag. Add the chicken pieces and shake to coat. Reserve 1/2 cup of the flour. In a large skillet, heat 1 1/2 inches of oil until very hot (375-400 degrees). Fry the chicken pieces until the crust is brown on both sides and the meat is cooked (8-10 minutes per side). Drain the chicken on paper towels and carefully pour the hot oil into a glass measuring cup, leaving the browned particles in the pan. Scrape the pan bottom to loosen any stuck particles and return 1/2 cup oil to the pan.
Place the pan over high heat. Using a long-handled metal whisk, gradually stir in the reserved 1/2 cup flour. Cook, whisking constantly, until the roux is dark red-brown (about 3 1/2-4 minutes). Be careful not to let it scorch or splash on your skin. Remove from heat and immediately add the chopped vegetables, stirring constantly until the roux stops browning. Return the pan to low heat and cook until vegetables are soft, stirring constantly.
Meanwhile, place the chicken broth in a 6 quart pot and bring to a boil. Add the roux mixture by spoonfuls, stirring until dissolved between each addition. Reduce heat to a simmer and add the sausage and garlic. Simmer uncovered for 45 minutes. While the gumbo is simmering, bone the cooked chicken and cut into 1/2 inch dice. When the gumbo is cooked, stir in the chicken and adjust seasoning. Serve immediately with rice. 6 servings.