In a large nonstick skillet, melt 1 tablespoon of the butter over medium-high heat. Place the figs cut-side down and sauté until they are golden around the edges and pillow-soft to the touch, about 2 minutes. Turn and cook for 1 minute more.
Transfer to a plate and let cool slightly. Wipe out the skillet with a paper towel and set aside.
To assemble: Butter one side of each slice of bread with the remaining 2 tablespoons butter. Place 4 slices on your work surface, buttered side down. Spread the cheese evenly over the 4 slices, followed by the figs. Place the remaining 4 bread slices on top, buttered side up.
STOVETOP METHOD: Heat the same skillet over medium-high heat for 2 minutes. Put the sandwiches in the skillet (in batches if necessary), cover, and cook for 2 minutes, or until the undersides are golden brown and the cheese has begun to melt. Uncover, and turn the sandwiches with a spatula, pressing firmly to flatten them slightly. Cook for 1 minute, or until the undersides are golden brown. Turn the sandwiches again, press with the spatula, and cook for 30 seconds, or until the cheese has melted completely. Serve immediately.
SANDWICH MAKER METHOD: Preheat the sandwich maker. Follow directions for sandwich assembly, and cook according to the manufacturer's instructions.
GAS GRILL METHOD: Brush the grill rack with oil and preheat the grill to medium-high. Follow directions for sandwich assembly. Put the sandwiches on the grill and follow directions for the stovetop method.
VARIATION: If you're interested in adding an herbaceous note to this sandwich, take 1 tablespoon fresh rosemary (or 1 teaspoon dried rosemary) chop it, and mix it into the cheese before spreading on the bread.