Whisk together the eggs and milk in a medium bowl. Set aside.
Add about ½ tablespoon butter to a large sauté pan. Heat to medium, then add two slices of bread and heat until toasted. Add another ½ tablespoon of butter, flip the bread and toast the other side. Remove to a plate. Repeat with the remaining two slices of bread.
Heat another tablespoon of butter in the pan. Add the ham, bell pepper and onion, and sauté a couple of minutes, until softened. Add the egg mixture and sprinkle cheese on top. Season with salt and pepper. Let the eggs cook until lightly browned underneath. Take a rubber spatula and drag them around until cooked all the way through. Remove from heat.
Assemble the sandwiches, divide the egg mixture between two slices of bread. Add ketchup, if desired. Top with the other piece of bread, slice in half and enjoy!