Peel back chicken drumstick and breast skin without removing it all the way; make several deep cuts in the meat. Sprinkle 3 teaspoons Cajun seasoning into the cuts and onto the meat, then replace the skin. Whisk 1/2 cup olive oil and white wine together in a large bowl. Add chicken, spoon marinade over top, then cover and refrigerate for 1 hour.

Remove chicken from marinade and shake off excess; discard remaining marinade. Heat 1/4 cup olive oil in a large skillet over medium heat until shimmering. Add chicken, working in batches if necessary, and cook until browned, about 5 minutes per side. Transfer chicken to a large soup pot and pour broth over top. Bring to a boil, then reduce the heat and simmer until chicken is tender, about 20 minutes.

While the chicken is simmering, cook onion, bell pepper, and celery in a skillet with 3 tablespoons of olive oil, stirring occasionally, until onion is transparent and pepper has softened, about 8 minutes. Remove from the heat and set aside.

Mix flour and canola oil together in a nonstick pan to make a roux; stir until smooth and lump-free. Heat over medium-high heat, stirring constantly. After about 5 minutes, roux will start turning pale golden. Stir until the color becomes like milk chocolate, about 30-40 minutes. Remove from heat and stir until roux cools and stops cooking, about 5 more minutes.

Stir onion mixture and 1 cup boiling water into roux; heat over medium heat until it comes to a simmer. Whisk until thick, about 10 minutes. Add thickened roux to the chicken, stir together, and add remaining 9 cups boiling water. Season with Cajun seasoning and bring to a boil. Reduce heat and simmer for 1 hour.

Sprinkle garlic powder and cayenne pepper over sausage. Brown sausage in a skillet with 2 tablespoons of olive oil. Transfer sausage to the soup and simmer for 30 minutes. Remove chicken from the pot and take meat off bones; discard skin and bones. Return meat to the soup and simmer for 15 minutes. Skim off the layer of oil that forms on the top.

While gumbo is simmering, bring rice and 3 cups water to boil in a saucepan over high heat. Reduce heat, cover, and simmer until liquid is absorbed, about 20-25 minutes.