Get all ingredients ready. Once you are set up to cook, this dish goes together quickly.
Place vegetable oil and flour in a small saucepan over medium heat and whisk to combine. Cool, whisking constantly, until the mixture turns a dark caramel color and begins to smell like toasted almonds. This is roux, and will take about 15 minutes to produce a smooth paste that will not only thicken your gumbo, but lend a deep, rich color. Transfer the roux to a small bowl and let cool to room temperature. When the roux is cooled, drain the excess oil.
Place the chicken, onion, celery tops, and bay leaf in a large stockpot. Cover chicken with water, add the salt, and bring to a boil over medium-high heat. Reduce the heat and simmer until the chicken is tender and the water is infused with flavor, about 60 minutes. Remove the chicken and let it cool. Strain the stock and reserve. You'll need about 4 1/2 cups of stock.
When chicken is cool enough to handle, strip the meat from the bones and shred into bite-sized pieces.
Wipe out the stockpot and return it to medium-high heat. Warm the olive oil and add the green pepper, diced celery, and red onion. Cook, stirring occasionally, until the vegetables begin to soften and color lightly on the edges, 12-25 minutes. Combine the red, white, and black peppers, the file', and thyme together in a small bowl and sprinkle evenly over the vegetable mixture. Cook, stirring constantly, until the vegetables are well coated, about 8 minutes. When the spices are cooked, mix in the garlic, cooking for another 3 minutes.
Heat the stock in a saucepan over medium heat. Whisk 1/4 cup of the stock into the roux until it forms a smooth paste. Add it to the stockpot along with the shredded chicken and the remaining stock, stirring well to combine. Bring mixture to a boil, lower the heat, and simmer for 60 minutes, stirring occasionally.
While the gumbo is simmering, cook the bacon in a heavy skillet until the fat is rendered and the bacon is crisp, 5-8 minutes. Add the andouille sausage and stir to coat with the bacon drippings. Reserve.
After the gumbo has simmered for 60 minutes, add the tomato puree and bacon and sausage mixture. Take 1 cup of the hot gumbo liquid out and deglaze the bacon pan. Add these pan juice to the gumbo and continue simmering until the gumbo is slightly thickened, about another 30 minutes.
Stir in shrimp, cooking only until shrimp are pink, about 10 minutes.