Melt 2 Tbsp butter in a medium skillet over medium heat. Add onions then cook slowly, stirring occasionally, until caramelized and browned, about 30-40 minutes. Adjust heat and lower as needed if onions begin to brown too quickly. At the end, season onions with ½ tsp kosher salt, then stir and remove from the heat. Set aside.
Slice and spread a thin layer of brie on each bottom slice of baguette. Top with about 1 heaping teaspoon of caramelized onions.
Next, add about 1 tsp fig jam to each top slice of baguette. Place on top of the caramelized onions.
Meanwhile, heat a griddle, skillet, or stove top grill to medium heat. Melt 2 Tbsp butter, then add sandwiches and cook until golden brown on both sides and cheese has melted, about 3-4 minutes (depending on the thickness of the bread and cheese). While the first side is cooking, spread the top piece of bread from each sandwich with a little softened butter.
Serve immediately, either whole or sliced on a diagonal and served as half sandwiches.