Preheat the oven to 300°F. Line an 8x8-inch square baking pan with parchment paper and set it on a large-rimmed baking pan.
Whisk together the eggs, milk, salt, and pepper in a large bowl. Alternatively, use an immersion blender or high-speed blender to blend everything together until really smooth (my preferred method).
Pour the eggs into the prepared baking dish. Pour the hot water around the baking dish, into the rimmed baking sheet. The steam from the water bath creates an even, gentle heat for really fluffy eggs.
Bake in the preheated oven for 40-45 minutes. The eggs are done when the edges and top are mostly set, but it jiggles slightly when you nudge it.
While the eggs are cooking, toss the arugula with the olive oil and a pinch of salt. Use your hands to rub the olive oil into the arugula. Mix together the mayonnaise and Dijon mustard in a small bowl.
Carefully remove the eggs and sheet pan water bath from the oven. Take the eggs out of the water and carefully dump the water. Allow the eggs to cool slightly while you assemble the sandwich.
Preheat the oven to broil. Place the burger buns, inside-facing up, on the sheet pan. Brush each side with the melted butter. Broil for 2-3 minutes, to toast slightly, then add the shredded cheese to the top half of the burger buns and broil for another 1-2 minutes, until the cheese is melted.
Remove them from the oven and spread a generous amount of the mayo mixture onto the bottom buns.
Slice the eggs into 4 even squares or use a biscuit cutter to cut out round pieces of fluffy eggs. Place the eggs on the bottom buns with the mayo mixture. Top the eggs with the prepared arugula and top buns. Enjoy warm!