Score the duck breast skin (but not the meat) in a cross pattern. This helps the fat render and will give you a crispier skin. Place both skin-on breasts on a plate and set in refrigerator until ready to use. Salt it well on both sides, then let it sit on a cutting board while you prepare the gumbo.
Place a large Dutch oven over medium heat and add the vegetable oil. Whisk in flour, a little at a time and cook, whisking constantly for 15 to 20 minutes, until roux becomes smooth and thick and the color of a penny.
Stir in the onion, bell pepper, celery, and next 4 ingredients. Cook for about 5 minutes, stirring often, until vegetables are tender.
Add the minced garlic and cook another 2 minutes.
Add chicken broth, water, and Worcestershire. Bring to a boil. Reduce heat to low, then simmer 15 minutes.
Meanwhile, in a large skillet with 1 tablespoon of vegetable oil lay the duck breasts skin side down then turn the heat to medium-high and cook for about 5 minutes. Then flip them over and cook for another 3 minutes until an instant read meat thermometer shows it at 130 degrees F. Transfer the duck to a cutting board and let it rest, skin side up.
Add the andouille and diced tomatoes with juice to the gumbo, stir well to combine, and simmer another 10 minutes.
Add the shrimp and gumbo filé powder. Return heat to medium-high and cook 5 minutes or until shrimp turns pink.
Use a sharp knife to cut those beautiful crispy duck breasts into slices.
To serve, spoon ½ cup cooked white rice in the middle of serving bowl. Spoon 1 cup gumbo over the top, making sure each person gets some andouille and shrimp. Then top each with a few slices of duck breast.