Preheat the oven to 350 degrees F and line a baking sheet with foil. Whisk the eggs, milk, vanilla and salt in a shallow dish.
Combine the dried figs, dried apricots, nuts, granulated sugar and cinnamon in a small bowl and toss to combine.
Dip 4 of the bread slices in the egg mixture, then transfer to a clean surface. Top each slice with one-quarter of the fruit-nut mixture, then dip the remaining bread slices in the egg mixture and press on top to make 4 sandwiches.
Melt 1 tablespoon butter in a large skillet over medium-high heat. Working in batches, cook the sandwiches until golden brown, about 4 minutes per side, adding more butter to the skillet as needed. Transfer the sandwiches to the prepared baking sheet and bake until puffy and cooked through, 7 to 8 minutes.
Meanwhile, wipe out the skillet. Add the brown sugar, the remaining 4 tablespoons butter and 3 tablespoons water; cook over medium heat, stirring, until the sugar dissolves, about 3 minutes. Cut each sandwich in half and drizzle with the brown sugar sauce.