Melt butter in a large pot over medium-high heat. Add leeks and cook, stirring often, until softened but not browned, about 3 minutes. Add potatoes and broth. Increase heat slightly and bring to a boil. Reduce heat to medium-low and simmer until potatoes are tender, 10 to 15 minutes. Stir in peas; add 1 tsp. salt and 1/4 tsp. pepper. Return to a simmer and cook until peas and potatoes are very soft, about 10 minutes longer. Remove from heat and let cool.
Working in batches, carefully purée soup in a blender until very smooth. Strain through a coarse sieve, pressing hard on solids to extract as much liquid as possible. Discard solids. Cover and refrigerate until completely chilled, at least 2 hours. (Soup can be prepared to this point up to 1 day ahead. Keep covered and chilled.)
Whisk sour cream and pesto into chilled soup. (If soup is too thick, add milk or water a tablespoonful at a time to thin.) Season with salt and pepper.
