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Prepare oven and baking dish: Preheat oven to 400°F. Grease an 11- x 7-inch baking dish with 1 tablespoon of the butter.
Make sauce: Melt 2 tablespoons of the butter in a small saucepan over medium-high. Add flour; cook, whisking constantly, 1 minute. Slowly whisk in milk; bring to a boil, whisking constantly. Reduce heat to medium; cook, whisking constantly, until thickened, 4 to 5 minutes. Remove from heat; stir in salt and nutmeg. Set aside.
Make honey-mustard mixture: Stir together mustard and honey in a small bowl; set aside. Microwave remaining 1 tablespoon butter in a small microwavable bowl on HIGH until melted, about 20 seconds.
Build sliders: Arrange bottom roll halves in prepared baking dish; spread evenly with honey-mustard. Top evenly with ham, milk sauce, and cheese. Replace top roll halves; brush tops evenly with melted butter, and sprinkle with parsley. Cover with aluminum foil.
Bake sliders: Bake in preheated oven 8 minutes. Remove and discard foil. Continue baking until cheese is melted and rolls are golden, about 5 minutes. Serve immediately.