Step 1: Season the chicken. In a bowl, toss the chicken thighs with the Cajun seasoning, ensuring they are evenly coated.
Step 2: Sear the chicken. Heat the avocado oil in a large skillet or Dutch oven over medium-high heat. Add the chicken in batches, being careful not to overcrowd the pan. Sear the chicken for 3-5 minutes per side, until browned. Remove the chicken from the skillet and set aside.
Step 3: Make the roux. In the same skillet, reduce the heat to medium and melt the butter. Add the flour and whisk constantly until the mixture forms a smooth paste. Continue to cook, stirring constantly, for 10-15 minutes, or until the roux turns a medium-brown peanut butter color. Be careful not to burn the roux.
Step 4: Sauté the vegetables. Add the chopped onion, bell peppers, and celery to the skillet with the roux. Sauté for about 10 minutes, until the vegetables are softened and fragrant, stirring frequently to prevent sticking. Add the minced garlic and cook for 1 minute more, stirring constantly.
Step 5: Simmer the étouffée. Slowly whisk the warmed chicken broth into the skillet with the roux and vegetables, ensuring there are no lumps. Add the browning sauce (if using) and stir to combine. Return the seared chicken to the skillet. Bring the mixture to a simmer, then reduce the heat to low, cover, and cook for 30 minutes, or until the chicken is cooked through and the sauce has thickened to your desired consistency, stirring occasionally.
Step 6: Finish and serve. Remove the chicken from the skillet and chop into bite-sized pieces (if not already). Return the chicken to the skillet. Stir in the chopped parsley and half of the chopped green onions. Taste and add more Cajun seasoning if needed. Serve the Chicken Étouffée hot over cooked white rice, garnished with the remaining green onions.