1. Place oil in an 8- to 10-quart pan over medium-high heat. Squeeze sausages from casings into pan; discard casings. Break sausage into bite-size chunks. Stir occasionally until lightly browned, about 5 minutes.
2. Add onion, bell pepper, and garlic. Stir over high heat until onion is limp, about 5 minutes.
3. Stir in chili powder, cumin seed, and oregano. Add broth, cover, and bring to a boil over high heat.
4. Meanwhile, peel squash and cut into 1/2-inch cubes. Add to broth. Return to a boil. Reduce heat, cover, and simmer until squash is tender when pierced, about 15 minutes, stirring occasionally.
5. Add hominy. If peas are frozen in a block, whack container against counter to separate, then pour peas into soup. Turn heat to high and bring soup to a boil, about 3 minutes.
6. Pour soup into a tureen and ladle into mugs or bowls. Add salt and pepper to taste.