Place a rack into lower half of oven and preheat oven to 400 degrees F (200 degrees C).
Arrange bottom halves of Hawaiian rolls on a baking sheet. Place a half slice each of ham, Swiss cheese, smoked turkey, and provolone cheese onto each roll bottom. Place top halves onto each bottom to make sandwiches.
Melt butter in a saucepan over low heat and stir in sugar, dried onion flakes, poppy seeds, and honey mustard until mixture is smoothly combined, creamy, and the sugar has dissolved, about 2 minutes; brush over the tops of each sandwich.
Bake sandwiches on the preheated oven's lower rack until tops are golden brown and fillings are hot, about 15 minutes.
