1. Thinly slice white parts of 3 scallions. Melt butter in a large saucepan over medium-high heat. Add sliced scallions and cook, stirring often, until softened, about 3 minutes. Add peas and mint. Pour in broth and bring to a simmer. Cook, stirring often, until peas are very tender, about 10 minutes. Season with salt and pepper.
2. Working in small batches, transfer soup to a food processor or blender. Puree until smooth. Pour into a glass bowl; cool for 15 minutes. Loosely cover bowl with plastic wrap and refrigerate until well chilled, at least 4 hours.
3. Before serving, stir yogurt into soup. Thinly slice all but 1 inch of green parts of remaining scallions. Season soup with salt and pepper, if desired. Pour into chilled shot glasses; garnish with scallions or mint, if desired. Serve cold.