In a large, heavy pot heated over medium-high heat, warm the olive oil. Add the chicken in a single layer, working in batches if necessary, and sprinkle with 1 teaspoon of the salt (divide the salt between the batches if necessary). Cook, stirring now and then, until browned all over, about 15 minutes. Add the onion, bell peppers, garlic, cumin, chile powder, and oregano and cook, stirring now and then, until the vegetables begin to soften and brown in spots, about 10 minutes. Add the tomatoes with their juice, the black-eyed peas with their liquid, and the remaining 1 teaspoon salt. Bring the mixture to a boil and then immediately lower the heat to a gentle simmer.
Cover with the lid slightly ajar to let some steam escape. Cook, stirring now and then and breaking up the tomatoes as you stir, until all of the flavors have melded and the chicken is very tender, about 1 hour. (If you’re using thighs, the meat will begin to shred.) Add the pickled jalapeño liquid, then taste the chili and add more jalapeño liquid and/or salt if needed. (If you can, let the chili cool down, then cover and refrigerate it overnight and reheat it the next day. The flavor will be even better.)
If using, set out the hot sauce, sour cream, cheddar cheese, avocado, cilantro, pickled jalapeños, and scallions. Serve the chili piping hot.