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Whisk together eggs, parmesan cheese, and milk in a small bowl.
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Melt 1 tablespoon butter in a large skillet over medium heat. Add egg mixture and use a spatula to pull eggs to the center and tilt pan to fill edges in the same manner as making an omelette. Cook until just set. Transfer to a plate and cut in half.
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Spread remaining butter on one side of each bread slice. Turn slices over and spread a thin layer of whole grain mustard on each slice.
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Place one slide of Havarti cheese on mustard side of each slice of bread.
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Divide prosciutto in half and place on two slices.
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Place egg halves on top of proscuitto.
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Place the two other slices of bread on top of egg, cheese side down and butter side up.
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Place sandwiches in the skillet over medium heat.
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Cook until one side is golden brown. Flip sandwich over and cook remaining side until golden brown.
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Placing a lid over the pan while grilling the sandwich can help with melting the cheese without burning the bread. I cover it with a lid about half-way through grilling.