Preheat the oven to 375 degrees F.
Lay out the bread slices. Divide the ham among them, making sure it doesn't extend over the edges of the bread. Place the cheese over the ham. If the cheese is larger than the bread, bend it over to fit.
Heat 2 large ovenproof nonstick pans or griddles over medium heat. (If you have only 1 large pan, make 2 sandwiches and keep them warm in the oven while you make the second batch.) Add 1 tablespoon of the butter to each pan. When it has melted, add half the bread, cheese-side up to each pan and cook for 1 to 2 minutes, or until the bottoms are golden brown. Transfer the pans to the oven for 2 to 3 minutes to melt the cheese.
Meanwhile, melt the remaining 1 tablespoon of butter in a large ovenproof skillet and fry the eggs. Cook the eggs until the bottoms are set, then place the skillet in the oven for a minute to set the top of the whites. (We cook the eggs in 4 to 5-inch individual skillets.)
When the cheese is melted, remove the sandwiches from the oven. Place 2 slices together to make each sandwich and put each sandwich on a serving plate. Place an egg on top of each sandwich. Pour about 1/4 cup of the sauce over the white of each egg, leaving the yolk uncovered. Grind black pepper over each egg and garnish the eggs with a diagonal sprinkling of chopped parsley. Serve with frites, if desired.
Combine the water and yeast in a small bowl. Let stand for 10 minutes, then stir until the yeast is completely dissolved. Set aside.
Sift together the flours, sugar, and salt into the bowl of a stand mixer fitted with the dough hook. Add the eggs and beat for 1 minute at low speed, scraping down the sides of the bowl with a rubber spatula as needed. Slowly add the dissolved yeast and continue beating at low speed for 5 minutes. Stop the machine, scrape any dough off the hook, and beat for another 5 minutes.
Add about one-quarter of the butter cubes at a time, beating for about 1 minute after each addition. Once all the butter has been added, beat for 10 minutes more.
Place the dough in a large floured bowl and cover with plastic wrap. Set aside in a warm place until doubled in size, about 3 hours.
Turn the dough out onto a generously floured work surface and gently work the air bubbles out by folding the dough over several times while lightly pressing down on it.
Return the dough to the bowl, cover with plastic wrap, and refrigerate overnight.
The dough is now ready to shape or use in another recipe. Generously butter two 8 1/2 by 4 1/2-inch loaf pans. Turn the dough out onto a floured work surface. With floured hands, divide the dough in half and shape it into two rectangles that fit in the loaf pans. Place the dough in the pans.
Let the dough rise uncovered in a warm place until it is about 1/2-inch above the top of the pans, about 3 hours.
Preheat the oven to 350 degrees F.
Bake the brioche in the center of the oven until it is well browned on top and sounds hollow when tapped on the bottom, 35 to 40 minutes. Remove the brioche from the oven and immediately turn out onto a wire rack.
If serving immediately, let the breads cool for 10 minutes, then slice. If serving within a few hours, wrap the hot bread in aluminum foil and set aside at room temperature until ready to use. To freeze, wrap the hot bread in foil and promptly freeze. The bread can be kept frozen for up to 1 month; when ready to use, reheat (without thawing and still wrapped in the foil) in a 250 degree F oven until heated through, 20 to 25 minutes.
If using the brioche for croutons, let sit at room temperature uncovered to dry for a day.
Melt the butter in a medium heavy saucepan set on a diffuser over medium heat. Add the onion and a pinch of salt and cook slowly, stirring occasionally, for 2 to 3 minutes, or until the onion is translucent. Sprinkle in the flour and cook for about 3 minutes, stirring constantly so that the roux doesn't burn or color. Whisking constantly, add the milk and cream and whisk until fully incorporated. Bring to a simmer, whisking, then add the bay leaf, peppercorns, and cloves. Move the pan to 1 side of the diffuser, away from direct heat to avoid scorching, and bring back to a gentle simmer. Adjust the heat to maintain a gentle simmer and cook, whisking occasionally, reaching into the corners of the pan, for about 30 minutes. (If the sauce does begin to scorch, pour it into a clean pan--don't scrape the bottom of the pan--and continue.)
Remove the sauce from the heat and season to taste with salt, a grating of nutmeg, and a pinch of white pepper. Strain the sauce, add the cheese, and whisk to melt. Use immediately, or place in a storage container, press a piece of plastic wrap against the surface to keep a skin from forming, and refrigerate for up to a week. If the sauce is too thick after refrigeration, it can be thinned with a little heavy cream.
Jean-Louis Palladin was a close friend and one of the greatest chefs I've ever known. And he made some of the best brioche I've ever tasted. This is his recipe. Start it a day before you want to make it, as the dough has to rest overnight. Mornay, a roux-thickened white sauce, is a luxurious cheese sauce that we use for gratineed scallops, macaroni and cheese, croque-monsieurs and -madames, and crepes. VARIATION: For 3 cups sauce, use 4 tablespoons (2 ounces) butter, 2/3 cup diced onion, 1/4 cup flour, 22/3 cups milk, 11/3 cups cream, 4 peppercorns, 4 cloves, and 1/2 cup cheese. Make the sauce in a large saucepan.