Melt butter in a pot over medium heat.

Add thinly sliced leeks and celery to the pot.

Season with salt and pepper.

Cook, stirring frequently, until leeks are tender (about 5 minutes).

Add diced potatoes, broth, and 1 cup of water to the pot.

Cover the pot and bring the mixture to a boil over high heat.

Reduce heat to medium-low and simmer gently, covered, for about 15 minutes, or until potatoes are soft.

Remove the pot from heat.

Using a blender (in batches) or an immersion blender, carefully puree about half of the soup until smooth, leaving the other half chunky.

Return the pureed soup to the pot, if necessary.

Stir in milk.

Heat over medium heat, stirring frequently, until the soup is just steaming (do not let it boil).

Stir in freshly squeezed lemon juice.

Season to taste with additional salt and pepper, if needed.

Serve hot.