Cut chorizo into 1/4-inch-thick slices. In a 6- to 8-quart pan over medium-high heat, stir chorizo often until lightly browned, about 10 minutes. With a slotted spoon, transfer to towels to drain. Spoon out and discard all but 1 tablespoon fat in pan.
Add onion and garlic to pan and stir often until limp, about 5 minutes. Return chorizo to pan and add chicken and beef broth, potatoes, carrot, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Cover and bring to a boil over high heat; reduce heat and simmer, uncovered, until potatoes and carrot are tender when pierced, about 15 minutes.
Meanwhile, rinse and drain kale. Cut out and discard tough center ribs; chop leaves crosswise into 1/4-inch-wide slices. Add kale and cabbage to soup and stir occasionally until tender to bite, 8 to 10 minutes. Add salt and pepper to taste.