Makes about 10 cups, serving 8 to 10.
In a kettle cook the garlic, the onions, and the carrot in the oil over moderately low heat, stirring, until the vegetables are softened. Add the russet potatoes, peeled and cut into 1-inch pieces, the broth, and 4 cups water, bring the liquid to a boil, and simmer the mixture, covered, for 10 to 15 minutes, or until the potatoes are tender. While the potatoes are cooking, in a skillet cook the chorizo over moderate heat, stirring, until it is browned lightly and transfer it with a slotted spoon to paper towels to drain. With the slotted spoon transfer the cooked potatoes to a blender, add about 1 1/2 cups of the cooking liquid and purée the mixture until it is smooth. Stir the purée into the broth mixture, add the chorizo, the kale, the red potatoes, cut into 1-inch pieces, and salt and pepper to taste, and simmer the soup, covered, for 10 minutes, or until the potatoes are tender.