Finely chop onion, bell peppers, celery, green onions, and parsley while mincing garlic. Measure out all other ingredients carefully before starting. Season chicken thighs generously with cajun seasoning.
Heat a large skillet over medium-high heat and add avocado oil until shimmering. Carefully place chicken thighs in batches, searing each side for about 3-5 minutes until golden brown. Set aside.
In the same skillet, reduce heat to medium and melt butter until bubbling. Gradually whisk in flour, stirring constantly for about 10-15 minutes until roux reaches a medium brown peanut butter color.
Add chopped onions, bell peppers, celery, and half of the green onions into the roux mixture, stirring for about 10 minutes until vegetables soften. Add minced garlic during the last minute.
Slowly incorporate warmed chicken broth into the roux mixture by whisking continuously. Stir in browning sauce and reintroduce seared chicken. Simmer on low heat for about 30 minutes until desired thickness is achieved.
Remove chicken thighs from pot, chop into bite-sized pieces, and return them along with freshly chopped parsley. Taste and adjust seasoning if necessary. Serve over fluffy white rice, garnished with remaining green onions.