Season duck legs all over with salt and pepper.

Heat vegetable oil in a large skillet over high heat. Place duck legs, skin-side down, in hot oil and cook until browned, 2 to 5 minutes. Turn and cook until browned on the other side, 2 to 4 minutes. Transfer legs to a plate; pour rendered duck fat into a bowl.

Pour water into the skillet and bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Remove from the heat.

Place about 2 tablespoons duck fat into a Dutch oven; add olive oil and heat over medium heat until hot. Add onion and sauté until golden brown, 7 to 10 minutes. Add turmeric, cinnamon, and nutmeg; cook and stir until fragrant, 1 minute.

Pour chicken broth, pomegranate molasses, honey, and reserved water mixture from the skillet into the Dutch oven; bring to a simmer.

Meanwhile, grind walnuts to a fine powder in a food processor.

Cook and stir ground walnuts in a dry skillet over medium heat until fragrant, 2 to 3 minutes; stir into broth mixture.

Place duck legs into broth and press gently to submerge. Reduce the heat and simmer until duck legs are no longer pink at the bone and the juices run clear, 3 to 4 hours. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).