Heat oil in a large saucepan or Dutch oven over medium-high heat: add chicken and cook until browned, approximately 2 minutes per side. Remove chicken from the saucepan and set aside.
Add onions and garlic to saucepan and sauté over medium-low heat until golden, about 2 to 3 minutes. Stir in tomatillos and jalapeño peppers. Add broth and bring to a boil; reduce heat to low, cover, and simmer for about 15 minutes.
Puree soup in batches in a blender or food processor. Return to pot, and reheat. Add hot pepper sauce and cayenne pepper to taste.
