Pulse gizzards in food processor, then pulse livers.
Heat oil in a large saucepan. Saute onion and gizzards over medium heat, stirring constantly, until meat begins to brown (about 5 minutes). Add livers and garlic and 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, then reduce heat to simmer. Cover and cook about 15 minutes, until rice has absorbed liquid. Sprinkle with green onion.
Smoke on BBQ pit with wood for an hour or two depending on how deep you want the smoked flavor to be adding water as necessary.