Place a large soup pot over medium heat and coat the bottom with olive oil. Add the onion and garlic. Cook until the onions are soft, about 4 minutes.
Add the chestnuts and potatoes and cook until all of the ingredients are beginning to brown, about 5 minutes. Stir in the herbs and spices and cook until it's very aromatic, about 30 seconds. Then add the stock, bring to a boil, reduce the heat to a low simmer, and cover. Simmer the soup until the potatoes are tender, about 20 to 30 minutes.
Use an immersion blender to purée the soup until it's as smooth as possible. Then place a fine mesh strainer over a large bowl, or another pot, and strain the soup. Use a spoon to stir the soup in the strainer to help it through. Tiny bits of the chestnuts will be left in the strainer, and a lovely, thick soup will be below.
Season the soup to taste with salt and pepper. Serve each portion with a bit of Asiago cheese and chives on top.