Set a large pot over medium heat and add the chorizo and olive oil. Cook, stirring occasionally, until the chorizo is crisp and the oil is red and infused, about 2 minutes. Remove the chorizo to a paper towel-lined plate and reserve; leave the oil in the pot.
Add the chicken broth, onion, chicken thighs, garlic, bay leaf, 1 1/2 teaspoons salt and 4 cups water to the pot. Bring to a boil and reduce to a simmer. Cook, partially covered, for 15 minutes. Add the potatoes and simmer until both the chicken and potatoes are tender when pierced with a knife, 10 to 15 minutes more. Remove the chicken to a bowl.
Mash the potatoes coarsely with a potato masher to thicken the consistency of the soup. Discard the bay leaf. When the chicken is cool enough to handle, shred the meat and set aside.
Layer the kale leaves to make a stack. Cut the leaves in half lengthwise, then cut into 1/2-inch-wide pieces (or tear into 1/2-inch pieces). Add the kale and 1/4 teaspoon black pepper to the soup, return to a simmer and cook, uncovered, until the leaves are tender, about 5 minutes. Add the shredded chicken to the soup and return to a simmer.
Divide the soup among 6 bowls and serve topped with several pieces of the reserved chorizo.
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