Melt butter in a large heavy pot over medium heat. Add onion and cook, stirring often, until softened but not browned, 6–8 minutes. Add 1 1/2 teaspoons salt and 2 cups water, stir to combine, and bring to a boil. Add peas and cook, stirring occasionally, until just tender, about 2 minutes.
Remove pot from heat and stir in yoghurt. Purée soup in a blender or in the pot using an immersion blender, thinning with water if the soup is too thick, until smooth. Season with salt and pepper and let cool to room temperature. Transfer to a resealable container, cover, and chill at least 2 hours or up to overnight.
Divide soup among bowls and top each with a swirl of Herbed Olive Oil.