In medium size saucepan bring water to a boil and add split peas. Reduce heat, cover, and simmer peas until soft, approximately 20 minutes. Drain and set aside.
Rinse saucepan. Using the same saucepan, melt butter on medium heat. Add shallots and cook until softened (not browned) for a couple of minutes.
Add broth and mint sprigs to the shallots, increase heat and bring to a boil. Add drained split peas and salt and pepper to taste. Reduce heat and simmer, stirring occasionally for approximately 5 minutes while flavors blend.
Discard the mint and puree the soup in a large blender until velvety smooth. Be careful the mixture will be hot.
Blend in the cream.
Salt and pepper to taste.
Chill to serve cold.
Blend sour cream and half & half together to garnish, adding fresh mint strips if you desire.