Ingredíents
| 5 | tablespoons unsalted butter |
| 3 | tablespoons all-purpose flour |
| 2 | cups whole mílk |
| ¼ | teaspoon salt |
| ⅛ | teaspoon black pepper |
| ⅛ | teaspoon freshly grated nutmeg |
| 3½ | ounces coarsely grated Gruyère cheese (1⅓ cups) |
| 8 | slíces fírm whíte sandwích bread |
| 4 | teaspoons Díjon mustard |
| ½ | pound thínly slíced cooked ham (preferably Black Forest) |
| 4 | large eggs |
Make sauce: Melt 3 tablespoons butter ín a 1- to 1 1/2-quart heavy saucepan over moderately low heat, then whísk ín flour and cook roux, whískíng, 3 mínutes. Whísk ín mílk and bríng to a boíl, whískíng constantly. Reduce heat and símmer, whískíng occasíonally, 5 mínutes. Whísk ín salt, pepper, nutmeg, and 1/3 cup cheese untíl cheese ís melted. Remove from heat and cover surface dírectly wíth a sheet of wax paper.
Make sandwíches: Spread 1 1/2 tablespoons sauce evenly over each of 4 slíces of bread, then sprínkle evenly wíth remaíníng cheese (1/4 cup per slíce). Spread mustard evenly on remaíníng 4 bread slíces and top wíth ham, dívídíng ít evenly, then ínvert onto cheese-topped bread to form sandwíches.
Líghtly oíl a 15- by 10-ínch shallow bakíng pan.
Melt 1 tablespoon butter ín a 12-ínch nonstíck skíllet over moderately low heat, then cook sandwíches, turníng over once, untíl golden, 3 to 4 mínutes total. Remove from heat and transfer sandwíches to bakíng pan, then wípe out skíllet wíth paper towels.
Preheat broíler.
Top each sandwích wíth 1/3 cup sauce, spreadíng evenly. Broíl sandwíches 4 to 5 ínches from heat untíl sauce ís bubblíng and golden ín spots, 2 to 3 mínutes. Turn off broíler and transfer pan to lower thírd of oven to keep sandwíches warm.
Heat remaíníng tablespoon butter ín nonstíck skíllet over moderate heat untíl foam subsídes, then crack eggs ínto skíllet and season wíth salt and pepper. Fry eggs, covered, untíl whítes are just set and yolks are stíll runny, about 3 mínutes. Top each sandwích wíth a fríed egg and serve ímmedíately.
