Prepare the Ham. In a large skillet over medium-high heat, sear the ham until browned and fully heated through. Remove from the skillet and set aside.
Scramble the Eggs. In a bowl, whisk together the eggs, milk, salt, and pepper. In the same skillet used for the ham, melt 1 tablespoon of butter over medium heat. Pour in the egg mixture and cook, stirring gently, until the eggs are just set and still slightly runny. Remove from heat.
Assemble the Sliders. Preheat your oven to 350°F. Slice the slider buns in half horizontally. Place the bottom halves in a baking dish. Layer the cooked ham on the buns, followed by the scrambled eggs. Sprinkle the shredded cheddar cheese evenly over the eggs. Place the top halves of the slider buns on top.
Bake. Melt the remaining 1 tablespoon of butter and brush it over the tops of the buns. Cover the baking dish with aluminum foil and bake in the preheated oven for 10-15 minutes, until the cheese is melted and the sliders are heated through.
Serve. Remove from the oven and let the sliders cool slightly before serving. They go perfectly with a side of fresh fruit, hash browns, or a simple green salad.