If you're not into egg salad, try pairing this soup with chicken salad or grilled cheese sandwiches. During cooler months, you can make this soup with frozen peas.
Make the soup: In a medium pot, melt butter over medium-high. Add onion and saute until translucent, 8 minutes. Add garlic and cook until fragrant, 1 minute. Add 1 cup water and simmer until onions are soft, 8 to 10 minutes. Add peas and cook until bright green, 4 minutes; season with salt and pepper. Transfer mixture to a blender and add buttermilk. Blend until smooth, then pour through a fine-mesh sieve into a large bowl, pressing on solids; discard solids. Refrigerate until cold, 1 hour, then season with salt and pepper.
Make the egg salad: Remove yolks from 3 eggs and discard. Coarsely chop eggs and mix with mayonnaise, mustard, and celery; season with salt and pepper. Make sandwiches with bread, egg salad, and arugula. Serve with chilled soup.
A pound of English peas in the pod yields about 1 cup shelled peas.