Place two slices of bread on baking sheet, buttered side down; top with half of the sliced cheese and ham, dividing evenly; sprinkle with the dates.
Top with remaining cheese and the two remaining bread slices; buttered side up. Refrigerate for 30 minutes (or up to 8 hours covered and wrapped in plastic; peeling plastic wrap away before continuing).
Heat panini press and cook sandwiches according to manufacturer's directions. (Instead of using a panini press you can heat a heavy skilit on medium high heat; reduce heat to medium low and place sandwiches on skillet; place another skillet on top of sandwiches; place a weight on top of skillet and cook sandwiches until golden brown and cheese melts (3-5 minutes per side).