How to Make It
In a saucepan, melt the butter. Add the yellow squash and bay leaves and cook over moderate heat for 8 minutes. Add the broth and water and bring to a boil. Simmer just until the squash is tender.
Meanwhile, preheat a grill pan. Rub the scallions with 1 teaspoon of the oil and grill over high heat, turning occasionally, until lightly charred, 5 minutes. Let cool, then cut into 1/2-inch pieces. Transfer to a bowl.
Transfer the squash and 1/2 cup of the broth to a blender and discard the bay leaves. Puree the soup and blend in the buttermilk. Season with salt and pepper. Keep warm or refrigerate until cold.
Heat a nonstick skillet over moderate heat. Spoon twelve 2-inch-wide mounds of the Manchego into the skillet; cook over moderate heat until browned on the bottom. Off the heat, flip the cheese crisps. Return to the heat and cook until browned. Transfer the crisps to a rack and let cool.
Add the tomatoes, tarragon, parsley, cucumber, lemon juice and the remaining 1 tablespoon of oil to the scallions and toss. Season the salad with salt and pepper. Ladle the soup into bowls and garnish with the scallion salad. Serve the soup warm or chilled with the cheese crisps.