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Heat oven to 350°F. Spread cornbread pieces in single layer on baking sheet. Bake, stirring every 5 minutes, until bread is dry but not browned, about 20 minutes. Transfer to large bowl; set aside to cool.
Meanwhile, heat olive oil in large non-stick skillet over medium-high heat. Add chorizo; cook, stirring occasionally, until light golden brown, about 4 minutes. Add onions and poblanos to pan. Cook, stirring occasionally, until vegetables soften, about 10 minutes. Stir in garlic and oregano, cooking until fragrant and well-combined, about 30 seconds more. Remove skillet from heat; stir in 2 tbsp. cilantro. Transfer chorizo mixture to bowl with corn bread, mix until well distributed. Mix chicken bouillon mixture with eggs until well combined. Pour chicken bouillon mixture over cornbread mixture. Using hands, mix and knead cornbread until well saturated and combined.
Transfer cornbread mixture to 2-quart oval casserole dish; cover tightly with foil. Bake until stuffing is puffed and heated through, about 1 hour. Uncover casserole dish. Bake until top is slightly toasted, about 10 minutes more. Sprinkle with remaining cilantro before serving.