Heat the olive oil in a medium saucepan over medium heat. Add the onion and cook until translucent, about 10 minutes. Reduce the heat if the onion begins to brown.
Add the tomatillos, garlic, and jalapeno and cook for 5 minutes. Raise the heat to medium-high, add the chicken stock, cumin, and cilantro, and cook 10 minutes more. Remove from the heat and cool.
Pour the mixture into the bowl of a food processer and puree until smooth. Add the buttermilk, salt, and pepper and pulse to combine. Transfer to a bowl and chill in the refrigerator.
Ladle the soup among 4 bowls and garnish each with a cilantro sprig and the cumin.