Pulse gizzards in a food processor, then pulse livers.

Heat oil in a large saucepan. Sauté onion and gizzards over medium heat, stirring constantly, until meat begins to brown, about 5 minutes. Add livers and garlic; cook, stirring constantly, until brown, about 3 minutes.

Add rice and stir until coated with oil. Add broth, salt, pepper, and cayenne pepper. Bring to a boil; reduce heat to simmer, cover, and cook for about 20 minutes, or until rice has absorbed liquid. Sprinkle with green onion and serve.