French Onion Soup Monte Cristo Sandwich

**Caramelize the Onions:** Melt butter in a large skillet over medium-low heat. Add sliced onions and cook, stirring occasionally, until softened and deeply caramelized, about 25-30 minutes. Stir in brown sugar and balsamic vinegar. Cook for another minute. Deglaze the pan with beef broth, scraping up any browned bits from the bottom. Cook until the broth is mostly absorbed. Remove from heat and set aside.

**Prepare the Custard:** In a shallow dish, whisk together eggs, milk, nutmeg, salt, and pepper.

**Assemble the Sandwiches:** Divide the caramelized onions, Gruyere cheese, and ham evenly among 4 slices of brioche bread. Top with the remaining 4 slices of bread.

**Soak the Sandwiches:** Dip each sandwich into the custard mixture, ensuring both sides are well coated. Allow excess custard to drip off.

**Fry the Sandwiches:** Heat cooking oil in a large skillet or on a griddle over medium heat. Carefully place the soaked sandwiches into the hot oil and cook for 3-4 minutes per side, or until golden brown and cooked through. Be careful not to overcrowd the pan; you may need to cook in batches.

**Bake to Melt Cheese:** Place cooked sandwiches on a baking sheet and transfer to a preheated oven at 350°F (175°C) for 5-10 minutes, or until the Gruyere cheese is melted and bubbly. (Optional but recommended)

**Serve:** Cut each sandwich in half diagonally and serve immediately with berry jam for dipping.