In a saucepan, bring 1/2 cup water and the dates to a boil, then lower the heat and simmer for 5 minutes. Transfer the mixture to a food processor. Add 1/4 tsp. salt and process until smooth. Spread each ham slice with 1/2 tsp. date mixture.
In a large skillet, heat 1 tsp. butter over medium heat. Add 2 slices ham, date side down. Cook until golden-brown, about 1 minute. Spread the top of the slices with another 1/2 tsp. date mixture, flip the slices and cook for 1 minute more. Transfer to a plate. Repeat with the remaining butter, ham and some of the date mixture.
Spread the remaining date mixture on 10 bread slices. Top with the arugula and 2 pieces of ham. Spread the mustard on the remaining 10 bread slices and cover the sandwiches. Cut the sandwiches in half.