Spread the fig jam evenly over the cut side of the top half of the ciabatta.
In a small bowl, combine the crumbled blue cheese and softened butter and stir until smooth; if using goat cheese, you can skip blending and use it as-is.
Spread the cheese (or cheese-butter) mixture over the cut side of the bottom half of the ciabatta.
Arrange the cooked chicken evenly over the cheese layer and sprinkle with the freshly ground black pepper. Place the jam-covered top half of the bread jam-side down over the chicken.
Heat a large nonstick skillet over medium heat. Place the sandwich in the skillet, then set a heavy cast-iron skillet or other weight on top of the sandwich to press it.
Cook about 5 minutes, then carefully flip the sandwich and press again; cook until both sides are browned and the sandwich is heated through, about 5 minutes more (total about 10 minutes, adjusting as needed).
While the sandwich cooks, toss the arugula with the fresh lemon juice in a bowl until lightly dressed.
Remove the top half of the bread, mound the dressed arugula over the chicken, replace the top half, and cut the sandwich into 4 equal portions to serve.