Heat a large skillet over high heat. Once hot, lie the watermelon into the pan with the rind along the edge of the pan, leaving one half the the cooking surface free.
Crack eggs into the free space in the skillet. Top the eggs with 1/4 of the Parmesan cheese. Cook until the egg white begins to solidify on the bottom, 1 to 2 minutes. Begin folding the solid egg white over the translucent egg white. Continue cooking until the egg can be tilted with the edge of a spatula without the top part of it sliding away from the cooked egg white and is lightly crisped on the bottom, about 2 minutes more. Flip both the eggs and the watermelon pieces; sprinkle another 1/4 of the Parmesan cheese over the eggs and another 1/4 of the Parmesan over the watermelon pieces. Continue cooking until the egg whites are crisp on the underside, but the yolk remains soft when pressed lightly with the back of a spatula, 1 to 2 minutes. Remove eggs to a plate.
Continue cooking watermelon until darkened on the underside, about 1 minute more. Flip watermelon pieces again, top with remaining Parmesan cheese, and remove to the plate with the eggs.
Sear ham slices in any remaining liquid in the skillet just until lightly browned and heated through, 1 to 2 minutes per side. Transfer to the plate next to the watermelon and eggs.
You can also use a thick slice of ham or a few slices of bacon, if you prefer.
A nonstick pan is best, but it must be at least stainless steel. Do not use cast iron.
The eggs or watermelon can be salted to taste, but the Parmesan usually lends enough salt to the eggs, in my opinion, and the pan-grilled watermelon stands on its own without salt. Have this on its own or with toast or cereal.
The watermelon can be had with or without yolk; a bite of watermelon on a toast point with a drizzle of the parmesan egg yolk can be quite scrumptious.
Can be separated into two smaller servings, especially if toast or another side is added.