Season lobster tails with Tony Chachere's seasoning, garlic, and melted butter. Set aside.
Season diced chicken lightly with Cajun seasoning, salt, and pepper.
In a Dutch oven, melt butter over medium heat.
Add flour and whisk continuously until a dark peanut butter to chocolate-colored roux develops, approximately 20–30 minutes.
Add onion, celery, and bell pepper.
Cook for 5–7 minutes until softened.
Stir in garlic and cook for 1 minute.
Slowly whisk in chicken stock.
Add bay leaves, thyme, Cajun seasoning, salt, and pepper.
Simmer for 20 minutes.
In a skillet, cook chicken until lightly browned.
Add chicken to the gumbo.
Brown andouille sausage and add to the gumbo.
Simmer an additional 30 minutes.
Bake or broil lobster tails until fully cooked.
Remove lobster meat from tails and chop a portion for the gumbo.
Stir chopped lobster meat into the gumbo during the final 5 minutes of cooking.
Divide cooked white rice between serving bowls.
Ladle gumbo over rice.
Top each bowl with lobster tail and sliced green onions.
Serve immediately.