Heat oil in a skillet over medium heat. Cook duck legs in hot oil, skin-side down, until browned and fat has rendered, about 10 minutes. Flip and cook until the other side is browned, 3 to 4 minutes. Transfer to a plate, leaving rendered fat in the skillet.

Whisk 1 cup flour into fat in the skillet, adding enough oil to make a thick and smooth roux. Reduce heat to medium-low and cook, stirring constantly, until roux turns a rich, reddish-brown color, about 40 minutes. Whisk in 2 more tablespoons flour and cook for 2 minutes.

Whisk in broth, 1 cup at a time, until incorporated. Remove from heat.

Cook sausage in a large Dutch oven over medium heat until browned, about 8 minutes. Stir in onion, peppers, celery, and chopped green onions; cook and stir until onion is translucent, about 10 minutes. Stir in black pepper, thyme, cayenne pepper, and bay leaf, followed by diced tomatoes until combined.

Place ham hock into the center of the Dutch oven. Pour roux mixture over ham hock along with enough water to cover. Place duck legs into mixture. Bring to a simmer, then reduce heat to low. Cover with a lid set at an angle to let steam out. Simmer slowly, stirring occasionally, until duck and ham hock meat are tender, about 4 hours. Skim as much fat as possible off the top as it simmers.

Transfer duck legs and ham hock to a large bowl and let cool. Stir pickled okra into gumbo. Pick meat from duck legs and ham hock and return meat to gumbo. Simmer gumbo for 45 more minutes.