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 20 minutes, until rice has absorbed liquid.