Heat a heavy pot or Dutch oven over high heat. Add the extra-virgin olive oil, a turn of the pan, and the meat and brown until crispy, 8 to 10 minutes. Add the chorizo and render the fat a bit, then add the onions, garlic and chiles and season with salt and lots of black pepper, to taste. Cook for 5 to 6 minutes more to soften the onions, then add the oregano, paprika, and chili powder. Once the onions are tender add the tomato paste and stir for 1 minute. Stir in the stock, lower the heat, and simmer for a few minutes to combine the flavors. Can be cooled and refrigerated for a make-ahead meal and reheated over medium-low heat with a little extra broth or water to reconstitute. The chili also freezes well.
Put the herbs, shallots, garlic, salt and pepper, to taste, vinegar and oil in food processor bowl. Pulse-process until finely chopped but still a loose sauce. Sauce can be made fresh or refrigerated for a few days in airtight container.
Transfer the hot chili to serving bowls and topped with Chimichurri sauce. Serve with lots of crusty warm bread. You can also top with finely chopped seeded tomatoes, if desired.