In a large saucepan, melt the butter, and add 4 tablespoons olive oil. Add the onions, celery, and carrots to the pan, and sauté until transparent and just beginning to brown, about 5 minutes. Add the garlic and the tarragon, and stir for another minute. Add the white wine, and let the mirepoix simmer for about 7 minutes. Add the broth, and simmer for an additional 15 minutes.
To the pan, add the broccoli, and cook until just fork-tender, about 7 minutes. Reduce the heat to low. Add the cheese, stirring until melted.
While the soup cools, char the scallions. Toss the scallions in the remaining ½ teaspoon oil, spread out onto a small baking sheet, and place under a broiler set to high. Rotate the pan as needed until the scallions are covered in blackened and blistered spots. Remove from broiler, let cool, and chop into pieces for garnish.
When the soup has cooled slightly, add the spinach, cream, and sherry. Blend with a hand blender or in a regular blender until smooth. Season to taste. Portion into bowls, top with celery leaves and charred scallion, and serve immediately.