Heat olive oil in a heavy, large pot over medium heat. Add the leeks and the garlic and stir. Add the potatoes and cook for 5 minutes, stirring often. Add the artichokes, stock, salt, and pepper and cook until the vegetables are tender, about 20 minutes.
Using a handheld immersion blender, or in a blender in batches*, puree the soup. Add the 2 tablespoons mascarpone and blend again to combine. In a small bowl, stir the remaining 1/3 cup mascarpone to soften. Ladle the soup into serving bowls. Dollop the top of each of the soups with a spoonful of the softened mascarpone cheese and top the cheese with chives.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.
Recipe courtesy of Giada De Laurentiis