Melt butter with oil in heavy large pot over medium-high heat. Add onions and saute until golden brown, about 20 minutes. Add wine; simmer until wine evaporates, about 4 minutes. Add tomato and saute 1 minute. Remove from heat. Transfer 1 cup onion mixture to blender. Add 1 cup chicken broth and puree until smooth. Return puree to pot. Add remaining 3 cups broth, chopped prosciutto, and thyme. Bring soup to boil. Reduce heat to medium-low, cover, and simmer 30 minutes to blend flavors.
Meanwhile, preheat broiler. Place 4 prosciutto slices on baking sheet. Sprinkle each with 1 tablespoon Parmesan cheese. Broil until cheese melts and bubbles and prosciutto is crispy, about 1 minute. Cut each slice into 6 pieces.
Ladle soup into bowls and top each serving with prosciutto pieces.