Heat oil in a large skillet over medium-high. Add onion and cook until just soft, about 5 minutes. Add turkey and season with salt. Cook, breaking up meat with the side of a spoon, until meat is cooked through, 4 minutes.
Stir in tomato paste and chili powder and cook 1 minute. Add broth and simmer, stirring occasionally, until thickened, about 2 minutes more.
Double up lettuce leaves. Divide meat mixture among leaves. Sprinkle with cheese, tomato, onion, avocado, and cilantro and serve.