In a large skillet, heat 2 tablespoons butter over medium heat. Add onions, sugar and salt; cook and stir until softened, about 15 minutes. Reduce heat to medium-low; cook until deep golden brown, 30-40 minutes, stirring occasionally.
Meanwhile, place figs, water and honey in a small saucepan. Bring to a boil; reduce heat. Simmer, uncovered, until liquid is almost evaporated, about 20 minutes.
Preheat panini maker or indoor electric grill. Layer 6 slices of bread with Asiago cheese, prosciutto, goat cheese, caramelized onions and figs; top with remaining bread. Spread outsides of sandwiches with remaining butter.
Cook sandwiches, covered, until bread is browned and cheese is melted, 4-5 minutes. If desired, drizzle with balsamic glaze.