For the meatloaf, preheat oven to 350℉. Lightly toast the breadcrumbs and transfer to a bowl. Add milk and soak the breadcrumbs.
Place meat in mixing bowl and season with salt and pepper. Squeeze excess milk from breadcrumbs and add breadcrumbs to bowl with egg, thyme, garlic and EVOO. Combine meatloaf, and be careful not to overmix.
Line a 9 x 13-inch quarter-sheet size baking sheet pan with parchment and plastic wrap, with plastic hanging over the ends. Spread meat into an even layer in baking sheet, corner to corner.
Top the meat with an even layer of Dijon mustard, then the grated cheese, cornichons and ham, stopping about 1½ inches or so from far edge of pan. You are rolling from the short edge, away from you as evenly and firmly as possible, using the plastic wrap to help lift meat up as you roll. Form a seam to seal the loaf when you get to the far side where toppings end. Smooth the loaf and twist wrap at ends to set loaf and chill for a few hours or overnight. (If you make the loaf ahead, bring to room temp and pat dry before roasting.)
For the sauce, heat a skillet or saucepot over medium heat. Melt butter, whisk in flour to combine, and when the roux bubbles, add milk while whisking, then season with salt, white pepper and nutmeg. Temper the egg yolk by adding a ladle of the hot milk sauce to it while mixing the egg yolk with a fork, then add back to sauce and remove from heat.
Preheat oven to 400℉. Roast meatloaf about 30 minutes on the parchment-lined baking sheet, with roll centered lengthwise down center of the sheet pan. Spread the room temperature bechamel sauce over top of the loaf and top with Parm and parsley. Bake meatloaf another 20 minutes to bubbly and lightly golden brown on top. Let the rolled meatloaf stand 10-15 minutes. Cut into 1½-inch-thick slices to serve.
For Madame preparation, heat a small nonstick skillet over medium heat. Melt a pat of butter, cook egg gently up or over, soft to medium hard, and season with salt and pepper. Slide egg on top of the meatloaf slice.
Or, for leftovers, reheat slices of meatloaf in nonstick skillet or cast-iron pan over medium to medium-high heat to crisp and warm through, then top with egg.