Prepare all of the ingredients. Preheat oven to 425deg.
In a sauce pan over medium heat, add butter for sauce. Once melted whisk in heavy cream. Cook for 3 minutes then add dijon, garlic and thyme. Cook for 2 mintues then add parmesan cheese. Once fully incorporated, reduce heat to medium-low and keep warm until ready to build sandwiches, stirring occasionally.
Lightly toast each piece of bread (approximately 2 minutes). Brush once side of each piece with melted butter. Using a cup or biscuit cutter, cut a center hole out of 6 slices of bread. (save the circle of bread).
On a parchment paper lined baking sheet, line up the 6 full slices of toasted bread. Place slice of provolone on each. Next add 2-3 pieces of ham and then ladle sauce over top. Top with hole punched bread slice. Sprinkle with parmesan and parsley. Place in oven for 10 minutes.
Remove from oven and break an egg into each bread whole. Increase oven heat to 500deg. Add circles of bread to baking sheet. Return to oven for 10 more minutes (adjust time for desired doneness of egg, whites should be cooked through and yolk should be runny).
Keyword cheese, croque madame, egg, egg recipes, grilled cheese, Ham