Set a skillet over medium-high heat. Add ham and warm on each side. Remove ham from skillet and set aside.
Add butter to the skillet. Crack the eggs into a mixing bowl and whisk together, forming a foam. When butter is just melted, add whisked eggs and mix with a spatula until fluffy and golden (take care not to brown the eggs). Remove skillet from the heat and set aside.
Prepare the everything bagels on a cutting board. Add 2 slices of warmed ham to each of the bottom bagel halves. Divide the cooked eggs into 2 separate portions and add each portion on top of the ham slices. Top the eggs with 2 slices of Cooper Sharp cheese for each sandwich. Top with the crowns of the everything bagels and serve.