Place2 bread slices on prepared baking sheet,buttered side down. Top bread slices withhalf of sliced cheese (1 1/2 ounces each); topcheese with ham, dividing equally.
Sprinkledates over. Top with remaining cheese,then remaining 2 bread slices, buttered sideup. Refrigerate 30 minutes. DO AHEAD: Canbe made 8 hours ahead. Cover with plasticwrap and keep chilled. Peel plastic fromsandwiches before continuing.
Heat panini press and cook sandwichesaccording to manufacturer’s instructions.(Alternatively, heat heavy large skillet overmedium-high heat. Reduce heat to mediumlow.
Place sandwiches in skillet.
Placeanother large skillet atop sandwiches; placeweight, such as large can of tomatoes, atopskillet. Cook sandwiches until golden brownand cheese melts, about 4 minutes per side.)