Heat large pot over medium heat. When pot is warm, add butter and allow to melt completely. Swirl butter around bottom of pot occasionally while melting to fully coat pot.
Once butter is melted, add green onions and holy trinity. Sauté vegetables, stirring occasionally, until vegetables have softened and onions are translucent.
Reduce heat under pot to low. Sprinkle flour into pot, then stir until flour is completely incorporated into melted butter and mixture is golden brown, approximately 2 minutes.
Once flour mixture is golden brown, pour in sherry and whisk mixture until thick paste-like consistency forms.
Continue whisking while adding tomato paste and cajun seasoning. Once ingredients are incorporated, slowly add seafood stock in small increments, whisking constantly, until all liquid is fully incorporated.
Increase heat under pot to medium and allow mixture to come to low, gentle boil. Once liquid begins to boil, reduce heat to low and simmer 20 minutes or until onions, bell peppers, and celery are very tender and liquid has reduced slightly.
Remove pot from heat. See next step for very smooth bisque texture, or skip next step to use bisque as-is.
Optional: For smoother bisque, insert immersion blender into pot and carefully blend liquid and vegetables until desired consistency is achieved. See Notes if using standard blender.
With pot off heat, pour in heavy cream and stir until cream is well incorporated. Return pot to low heat. Taste bisque and add salt and cayenne pepper or hot sauce as needed.
Add shrimp to pot. Cook, stirring occasionally, until shrimp are warmed through and tender, approximately 5 minutes. Be careful not to overcook shrimp.
Once shrimp are ready, ladle bisque into serving bowls. Top with chopped chives, hot sauce, and freshly cracked black pepper. Serve immediately with crusty bread and Louisiana beer if desired.