Pat chicken thighs dry with a paper towel, then season with salt and Cajun seasoning. Mix flour, paprika, cayenne, white pepper, granulated onion, and garlic powder in a bowl.
Dredge the seasoned chicken in the flour mixture on each side until coated. Shake off excess flour.
Heat oil in a cast-iron or skillet over medium-high heat. Then fry chicken on both sides until golden brown — 4 to 5 minutes per side or until browned.
Remove chicken from the skillet and place it on a plate. Add onions to the skillet, and saute for 2-3 minutes. Then add garlic and thyme and continue cooking for 3-5 more minutes until onions wilt, occasionally stirring to prevent it from burning. Remove onions and set them aside.
Add butter to the pan, and let it melt. Then throw in the seasoned flour left over from dredging the chicken and stir with a wooden spoon until it comes together and forms a paste.
Slowly whisk in the chicken broth, followed by the milk, ensuring there are no lumps. Stir and scrape the pan well until everything is incorporated. The mixture should thicken slightly in about 5 minutes. Return wilted onions to the skillet. Thoroughly mix and season with salt.
Add browned chicken with its juice to the skillet. Bring to a boil, then reduce heat to a simmer.
Simmer for 30-45 minutes, occasionally stirring to prevent it from sticking to the bottom of the pan. Adjust seasonings to taste.
Garnish with parsley. Serve with mashed potatoes.