In a mixer fitted with a paddle attachment, cream the butter until smooth and fluffy. Add the flour, cheese, and pepper and mix at low speed until blended. Form the dough into a disk, wrap it in plastic wrap, and refrigerate at least 2 hours. (The recipe can be made up to this point and kept refrigerated up to 2 days in advance.) Remove the dough from the refrigerator about 30 minutes before rolling out.
On well-floured parchment paper, roll out the dough into a rough rectangle, a little less than 1/4-inch thick. The dough will be somewhat delicate and may crack in places, but you can just push the edges together with your fingers. Carefully transfer the dough to the cookie sheet and prick evenly all over with a fork. Cut with a pizza cutter or a knife into 1 1/2-inch squares, making a grid.
Bake in a preheated 375 degree F oven, just until the crackers turn golden, 15 to 20 minutes. As they come out of the oven, run your pizza cutter or knife again along the grid to separate the crackers (the edges will have baked together a little bit in the oven). Let cool on the pan, then gently separate the crackers with your hands. They will be slightly fragile. Cut the figs in half and serve with the crackers.