Chop veggies: Place onion, celery, green bell pepper, and garlic into the bowl of a food processor; pulse until all vegetables are very finely chopped (or chop by hand). Sear the chicken: Pat chicken dry with paper towels. Season chicken with 1 teaspoon salt and 1 teaspoon pepper. Heat 2 tablespoons olive oil over medium-high heat in a large Dutch oven/soup pot. Once hot, add the chicken and sear until golden, about 2 minutes per side. Remove chicken to a plate but leave the drippings. Cook the sausage. Heat 1 tablespoon olive oil over medium-high heat (medium if your stove runs hot) in the chicken drippings. Add the sliced andouille sausage in a single layer. Sear each side until deeply golden. Using a slotted spoon, transfer to a paper towel lined plate to drain; set aside.
Make the roux: Reduce heat to low. To the same pot, don’t wipe out, add 3/4 cup neutral oil. Add the flour and whisk until combined. Increase heat to medium. Continue to cook, whisking constantly, until the roux is a rich dark brown color, like dark chocolate, about 20 minutes. I suggest turning the heat down to medium-low once you hit the chocolate brown stage then continuing to cook until dark brown. If the flour begins to stick at any point, reduce heat/and or add additional oil - you don't want it to burn!
Sauté the vegetables: Stir in the chopped vegetables and cook over medium heat for about 5-7 minutes, until the onions are tender.
Cook chicken: Add the seared chicken back to the pot followed by tomatoes, chicken bouillon, all spices, bay leaves and chicken broth. Partially cover, leaving a 1-inch gap opening and bring to a boil, then reduce to a simmer over medium-low until the chicken shreds easily with two forks, about 15 minutes.
Shred the chicken: Remove the chicken to a cutting board and shred when cool enough to handle. Meanwhile…
Simmer again: Reduce the heat to medium-low and gently simmer, UNCOVERED, for an additional 20 minutes, stirring occasionally.
Add chicken and sausage: Add the shredded chicken back to the pot along with the seared sausage. Cover and simmer an additional 15 minutes OR simmer uncovered if you would like the gumbo even thicker. (You can also start simmering uncovered, then cover when it reaches desired consistency.)
Add shrimp. Turn OFF the heat and stir in the shrimp (if using) gumbo filé powder, green onions, and chopped parsley. Stir well, cover, and rest for 20 minutes. Adjust to taste. Remove the bay leaves and season with salt (I add ½ teaspoon), pepper and or/cayenne pepper to taste. The gumbo should be thick but you can thin it with additional chicken broth if desired.
Serve. Serve over a bowl of cooked white rice, garnish with sliced green onions, and parsley. Serve with hot sauce or cayenne pepper on the side.