Heat oil in a large pot over medium-high heat. Add chicken and sausage; cook and stir until evenly browned, about 7 minutes. Add onion, red bell pepper, poblano pepper, jalapeno pepper, salt, and black pepper; cook, stirring often, until softened, 5 to 7 minutes.

Pour chicken broth into the pot; bring to a boil. Reduce heat to medium and stir in butter and arrowroot powder to thicken the stew.

Whisk eggs in a small bowl.

Increase heat to medium-high to bring stew to an active simmer. Remove from heat. Drizzle eggs in slowly, stirring with a wooden spoon until cooked through, about 1 minute. Stir in parsley.