Heat the oven to 350°F. Split the connected Hawaiian rolls horizontally and place the bottom slab in a parchment-lined 9-by-13-inch pan.
Layer on half the Swiss and the ham. Spread jam thinly over the ham, then add turkey, remaining Swiss and the top slab.
Beat egg with milk and melted butter. Brush a thin even layer over the roll tops, letting it settle into the surface without pouring excess into the pan; discard unused egg wash.
Cover loosely with foil and bake for 20 minutes. Uncover and bake for 8 to 10 minutes until the egg coating is set and the filling reaches 165°F.
Rest for 5 minutes, dust lightly with powdered sugar and cut into 12 sliders.