1. In a large, heavy stockpot, heat 1 tablespoon of the vegetable oil. Add the reserved shrimp shells and chicken wings, season with salt and pepper and cook over moderately high heat, stirring occasionally, until the chicken wings are browned, about 10 minutes. Add 1 tablespoon of the tomato paste and cook, stirring, until glossy, about 2 minutes. Slowly pour in the stock, stirring with a wooden spoon to release any browned bits on the bottom of the pot. Add the bay leaves and bring to a boil. Skim the surface of the stock, reduce the heat to low and simmer for 1 1/2 hours, skimming occasionally.
2. Add the chicken legs to the stock and simmer until they are cooked through, about 40 minutes. Strain the stock into a heatproof bowl and transfer the chicken legs to a large plate. When the legs are cool enough to handle, remove the meat and discard the skin and bones. Tear the meat into 2-inch pieces and set aside. Skim the fat from the surface of the stock and reserve the stock.
3. In a large enameled cast-iron casserole, stir the remaining 2/3 cup of vegetable oil with the flour until smooth. Cook the oil and flour over moderately low heat, stirring constantly, until a deep brown roux forms, about 25 minutes. Add the garlic, scallion whites, celery, onions, green peppers and finely chopped andouille sausage. Cook over low heat, stirring occasionally, until the vegetables soften, about 15 minutes. Add the thyme, cayenne, 1 teaspoon of salt and 1/4 teaspoon of pepper and cook, stirring, for 2 minutes. Add the remaining 1 tablespoon of tomato paste and cook, stirring, for 1 minute. Gradually whisk in the reserved stock. Simmer over low heat for 1 hour, skimming occasionally.
4. Add the okra and simmer until just tender, about 8 minutes. Add the reserved shredded chicken and the andouille rounds and simmer for 10 minutes. Season the gumbo with salt and pepper. Add the shrimp and simmer for 1 minute. Turn off the heat and let the gumbo stand for a few minutes. Stir in the filé powder and scallion greens and serve hot.
Make Ahead: The recipe can be prepared through Step 3 up to 3 days ahead and refrigerated. Reheat gently and proceed.
Serve With: White rice.
Wine Recommendation: Cool this spicy gumbo with an inexpensive, tangy, fruity white wine, such as the 1999 Eno Friulia Pinot Grigio, or a light Latin-American beer, such as Cerveza-Aguila.
Notes: Filé powder is made from ground, dried sassafras leaves. It acts as a thickener for stews and should be added at the very end to avoid over-thickening. It's available in the spice section of most supermarkets.