Finely chop the onion, green and red bell peppers, celery, green onions, and parsley. Mince the garlic. Season chicken thighs on both sides with Creole or Cajun seasoning.
Heat oil in a large skillet or Dutch oven over medium-high heat. Sear chicken in batches for 3–5 minutes per side until browned. Set aside.
Lower heat to medium. In the same pan, melt butter and whisk in flour. Cook, stirring constantly, for 10–15 minutes until roux reaches a medium brown, peanut butter color.
Add onion, bell peppers, celery, and the white parts of the green onions. Cook for 8–10 minutes until softened. Add garlic and sauté 1 more minute.
Slowly whisk in chicken broth until smooth. Stir in browning sauce. Return chicken to the pan and reduce heat to low. Simmer uncovered for 30 minutes, stirring occasionally, until thickened and chicken is cooked through.
Remove chicken, chop into bite-sized pieces, and return to the pot. Stir in parsley and adjust seasoning as needed. Serve hot over white rice and garnish with remaining green onions or parsley.