Heat olive oil in a 6-quart stockpot over medium heat. When hot, add onions, garlic, salt, and cayenne pepper. Saute until onions are soft and translucent, 4 to 5 minutes.

Add chicken stock and bring to a boil. Once stock is boiling, add broccoli and cook until fork-tender, about 5 minutes.

Remove soup from heat and let cool slightly, then puree soup mwith an immersion blender, adding cheese in three additions while blending. Adjust seasoning if necessary and serve hot.