French Onion Soup Dumplings with Parmesan Broth

**Prepare the Caramelized Onions:** Melt butter in a medium saucepan over medium heat. Add onions and cook, stirring occasionally, until softened and golden brown, about 25-30 minutes. Stir in brown sugar and sherry (if using) during the last 5 minutes. Remove from heat and let cool slightly.

**Make the Dumpling Dough:** In a large bowl, whisk together flour and salt. Cut in cold butter until the mixture resembles coarse crumbs. In a separate small bowl, whisk together egg and ice water. Gradually add the egg mixture to the flour mixture, mixing until a dough forms. Wrap the dough in plastic wrap and refrigerate for at least 30 minutes.

**Assemble the Dumplings:** On a lightly floured surface, roll out the dough to about 1/8 inch thickness. Use a small cookie cutter or glass to cut out circles. Place a small spoonful of caramelized onions and a pinch of Gruyere cheese in the center of each circle. Moisten the edges of the dough with water, fold the dough over to form a half-moon shape, and pinch the edges to seal tightly. Make sure no air is trapped inside.

**Prepare the Parmesan Broth:** In a large pot, combine vegetable broth, Parmesan cheese, bay leaf, and thyme sprigs. Bring to a simmer over medium heat. Simmer for 15 minutes to infuse the flavors.

**Cook the Dumplings:** Gently drop the dumplings into the simmering broth. Cook for 8-10 minutes, or until the dumplings float to the surface and are cooked through.

**Season and Serve:** Remove bay leaf and thyme sprigs from the broth. Season the broth with salt and pepper to taste. Ladle the soup into bowls and garnish with fresh parsley. Serve immediately.