Heat butter in a 4-quart saucepot over medium heat. Add leeks and a pinch of salt, and cook over low heat, stirring occasionally, until leeks are translucent but have not yet begun to brown, about 5 minutes. Add potatoes, corn, bay leaf, thyme, and chicken stock. Bring to a boil and then reduce to a simmer. Cook until potatoes are tender, about 15 minutes. Turn off heat and let soup cool slightly, about 8 minutes. Remove bay leaf and discard.
Using a standard or immersion blender, puree soup until smooth. If using a standard blender, puree in small batches to prevent spillage. Transfer to a clean pot or storage container. Soup can be refrigerated for up to 2 days or frozen for up to 2 months.
When ready to serve, reheat soup. Add milk and stir well; use additional milk or stock if soup is too thick; simmer for 5 minutes. Adjust seasoning to taste and serve immediately.