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Preheat oven to 350°F with racks in upper and lower thirds of oven. Butter a 3-qt shallow baking dish.
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Crumble corn bread into 1/2-inch pieces, spreading out in 1 layer in 2 large 4-sided sheet pans. Bake, stirring occasionally, until dy, about 20 minutes. Cool completely and transfer to a large bowl.
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Meanwhile, cook chorizo in oil in a 12-inch heavy skillet over medium-high heat, stirring occasionally, until crisp, 1 to 2 minutes. Add onions, celery, garlic, oregano, and 1 1/4 tsp salt and sauté until vegetables are softened, about 10 minutes. Add to corn bread.
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Whisk together broth and egg, then pour over stuffing and toss well. Transfer to baking dish and cover tightly with buttered foil. Bake in upper third of oven 1 hour. Remove foil and bake until top is golden, about 15 minutes more.