Place two 2-foot pieces of nonstick aluminum foil next to each other so the long sides are just touching. Pinch the edges together and then fold them over several times to seal and form one large piece of foil. Line a rimmed baking sheet with the foil and set aside.
Crack the eggs into a large bowl and whisk well. Add the scallions, 1 teaspoon salt and 2 tablespoons water and whisk to combine.
Split the whole pack of attached dinner rolls in half horizontally using a serrated knife. Set aside.
Put the prepared baking sheet on the grill and add 1 tablespoon of the butter. Cover the grill and let the butter melt, about 3 minutes. Spread the butter all over the baking sheet using a pastry brush and then pour on the egg mixture. Place the attached roll halves cut-side down and side by side on top of the eggs and cover the grill. Cook until the eggs are completely set, about 10 minutes.
Meanwhile, butter another baking sheet with the remaining 1 tablespoon butter. Using heatproof mitts, top the rolls with this baking sheet and invert the whole thing onto the grill. Remove the top baking sheet and peel back the foil. Top half the eggs with the cheese and the other half with the ham. Cover the grill and cook until the cheese is melted, about 5 minutes. Using a large metal hamburger spatula, flip the bottom half of the rolls onto the other half, so the rolls are top-side up. Serve with hot sauce.