In a large pot, heat the olive oil over medium heat. Add the diced onion, green bell pepper, and celery. Sauté for about 5 minutes until the vegetables are softened.

Stir in the minced garlic and sliced andouille sausage. Cook for an additional 5-7 minutes, stirring occasionally, until the sausage is browned.

Pour in the chicken broth and bring to a simmer. Add the cooked rice, Cajun seasoning, black pepper, and salt. Stir well to combine and let it simmer for 10 minutes.

Crack the eggs into the pot, one at a time, and gently stir to incorporate. Allow the eggs to poach in the broth for about 5 minutes, or until they are cooked to your liking.

Remove the pot from heat and stir in the chopped green onions and parsley.

Serve hot, garnished with additional green onions or parsley if desired.