Prep the chorizo: If using fresh chorizo, remove the casings and crumble. If using Spanish cured chorizo, slice or dice into small pieces.
Set aside.
Brown the chorizo: Heat a large pot over medium. Add chorizo and cook until browned and some fat renders, about 5–7 minutes. If the pot looks dry, add a splash of olive oil.
Use a slotted spoon to set the chorizo aside, leaving the fat in the pot.
Soften the veggies: Add onion, carrot, and celery to the pot. Cook, stirring, until softened, about 5 minutes. Add a pinch of salt to help them sweat.
Build the base: Stir in the garlic, tomato paste, smoked paprika, cumin, and red pepper flakes.
Cook 1–2 minutes until fragrant and the tomato paste darkens slightly.
Add potatoes and broth: Return the chorizo to the pot. Add potatoes, broth, and bay leaf. Bring to a boil, then reduce to a simmer.
Simmer until tender: Cook 15–20 minutes, or until the potatoes are fork-tender and the broth is flavorful.
Skim any excess fat from the surface if you like.
Add greens: Stir in the kale or spinach. Simmer 3–5 minutes until wilted and tender.
Finish and season: Remove the bay leaf. Stir in lemon juice or vinegar.
Taste and adjust with salt and pepper. If you want a slightly thicker texture, mash a few potatoes against the side of the pot and stir.
Serve: Ladle into bowls, top with chopped parsley, and serve with crusty bread. A drizzle of good olive oil on top is a nice touch.