Melt butter with olive oil in a large, heavy-bottomed saucepan over medium.
Add leeks; cook, stirring often, until softened, 5 to 8 minutes.
Add celery; cook, stirring often, until tender and deeply saturated in color, about 8 minutes.
Add potatoes; stir to coat potatoes in butter and oil. Reduce heat to low, and cook, stirring often, until potatoes are coated in a shiny, milky starch, about 5 minutes.
Stir in 7 cups water and chicken stock base; bring to a boil over high. Reduce heat to medium, and simmer, stirring occasionally, until potatoes are just tender and vegetables are softened, about 15 minutes. Remove from heat.
Using a ladle, remove 2 cups of cooking stock (liquid only), and reserve in a small bowl. Let the remaining soup stand at room temperature for 30 minutes.
Stir milk, cream, salt, pepper, and Italian seasoning into reserved stock. Stir in reserved stock, a little at a time, to reach desired consistency.
Place ¾ of the soup in a blend and puree, Add back into the reserved ¼ cup. If you are using an immersion blender to puree, remove ¼ of soup. Puree remaining soup in the pan. Add ¼ reserve back into pureed soup.
Spoon into bowls, top with parmesan cheese, drizzle of olive oil, chives and a few croutons.