Prepare the baking dish: Preheat the oven to 350°F. Lightly grease a 9-by-13-inch baking dish with cooking spray or butter.
Slice the rolls: Without separating the Hawaiian rolls, slice the entire slab horizontally with a serrated knife. Place the bottom half in the prepared baking dish and set the top half aside.
Cook the eggs: Whisk eggs, milk, salt, and pepper in a large bowl. Melt 1 tablespoon butter in a large nonstick skillet over medium-low heat. Add the egg mixture and gently scramble until the eggs are softly set but still slightly glossy.
Layer the sliders: Place 6 slices of cheddar cheese on the bottom rolls. Add the ham in an even layer. Spoon scrambled eggs over the ham, then sprinkle with green onions or chives if using. Add the remaining 6 slices of cheese.
Add the tops: Place the top half of the rolls over the cheese. Press gently to hold the sliders together.
Make the topping: In a small bowl, stir together melted butter, Dijon mustard, maple syrup, garlic powder, dried parsley, and optional everything bagel seasoning.
Bake: Brush the butter mixture over the tops and sides of the rolls. Cover loosely with foil and bake for 15 minutes. Remove foil and bake for another 8 to 10 minutes, until the cheese is melted, the tops are golden, and the center of the egg filling reaches 160°F.
Rest and serve: Let the sliders rest for 5 minutes. Slice between the rolls and serve warm.