Place a rack into the lower half of your oven and preheat oven to 400 degrees F (200 degrees C).

Arrange the bottom halves of the 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 the top halves onto each bottom to make sandwiches.

Melt the butter in a saucepan over low heat and stir in the sugar, dried onion flakes, poppy seeds, and honey mustard until the mixture is smoothly combined, creamy, and the sugar has dissolved, about 2 minutes. Brush over the tops of each sandwich.

Bake the sandwiches on the preheated oven's lower rack until the tops are golden brown and the fillings are hot, about 15 minutes.