• Pre-heat oven to 375 degrees.
• Place bread slices on cutting board or prep surface. Divide ham among them, being sure to trim it so it doesn't extend over the edges of the bread. Place cheese over ham, making sure it doesn't extend over the edges of the bread either.
• Heat griddle (or two oven-proof nonstick pans) over medium heat and add 2 Tbsp butter.
• Once butter has melted, place the bread cheese side up and cook for 1-2 minutes, until bottoms are golden brown.
• Transfer pans to the oven for 2-3 minutes, in order to melt the cheese.
• Melt the remaining 1 Tbsp butter in a separate large ovenproof skillet and fry the eggs. Cook until bottoms are set and then place skillet in the oven to set top of whites, about 1 minute.
• Once cheese on bread is melted, remove from oven and put two slices together to make sandwiches.
• Place sandwiches on separate plates, top with eggs, and pour ¼ cup mornay sauce over the white of each egg, leaving the yellow uncovered.
• Sprinkle black pepper over each egg and garnish with a sprinkling of fresh parsley.

Mornay Sauce

• Melt butter in a medium-sized heavy saucepan over medium heat.
• Add onion and a pinch of salt and cook slowly for 2-3 minutes, until onion is translucent, stirring occasionally.
• Sprinkle in flour and cook for about 3 minutes. Be sure to stir constantly, so roux doesn't burn.
• While continuing to whisk constantly, add milk and cream and whisk until fully incorporated.
• Bring to a simmer and add bay leaf, peppercorns, and cloves, while continuing to whisk.
• While keeping a gentle simmer, cook for about 30 minutes, whisking occasionally and being sure to reach corners of the pan.
• Remove pan from heat and season with salt, a pinch of nutmeg and pinch of pepper.
• Strain sauce and whisk in the cheese until it melts.
• If not using immediately, press plastic wrap against the surface of the sauce and refrigerate for up to a week. If necessary, sauce can be thinned with a little heavy cream.