Grease six 8-oz. ramekins; set aside. Preheat oven to 350°F.
In a large saucepan, bring 3 cups water to a boil. Add broccoli; cover and boil for 3 minutes. (I used the microwave -- Place broccoli in a microwave safe dish and add about a 1/2 cup water and cook on HIGH for 3 minutes). Drain and immediately place broccoli in ice water. Drain and pat dry.
In a large bowl, whisk 4 eggs, cream, cheese, parsley, salt, basil and cayenne; pour into prepared dishes. Break an additional egg into each dish; place broccoli around each egg.