Melt butter in a large pot over medium-high heat. Cook and stir leeks in the melted butter until tender, 5 to 7 minutes.
Stir potatoes and chicken broth into the leeks and bring to a boil; reduce heat to medium-low, cover the pot, and simmer until potatoes are tender, about 12 minutes.
Stir peas and tarragon into the soup; simmer until peas are tender, about 5 minutes.
Pour the soup into a blender, filling the pitcher no more than half full.
Hold down the lid of the blender with a towel and carefully start the blender, using a few quick pulses to get the soup moving before leaving it on to puree. Puree in batches until smooth and pour into a clean pot. Alternately, you can use a stick blender and puree the soup in the pot.
Pour milk into the pureed soup and bring to a simmer over medium heat until heated through. Season with salt and black pepper.