Trim fat from ham slices and place fat in large cast-iron skillet, setting ham aside. Cook fat over medium heat until it renders, about 3 minutes. (If there isn’t much rendered fat, add vegetable oil.)
Add ½ cup coffee to skillet. Add brown sugar and stir until melted. Place ham slices on top and cover skillet with lid. Cook over medium heat until wisps of steam come out from under lid, then uncover and cook ham until lightly browned.
Transfer ham to plate and keep warm. Discard any remaining pieces of fat. Add remaining ½ cup coffee. Increase heat to medium-high, and cook, stirring up from bottom, until gravy comes together and cooks down, about 2 minutes. Serve hot with ham slices, as well as biscuits for sopping.