Melt 3 tablespoons butter in a heavy saucepan
over low heat, then whisk in flour and cook
roux, whisking, 3 minutes. Whisk in milk and
bring to a boil, whisking constantly. Reduce
heat and simmer, whisking occasionally, 5
minutes. Whisk in salt, pepper, nutmeg, and 1/3
cup cheese until cheese is melted. Remove from
heat and cover surface directly with a sheet of
Spread sauce evenly over each of 4 slices of
bread, then sprinkle evenly with remaining
cheese (1/4 cup per slice). Spread mustard
evenly on remaining 4 bread slices and top with
ham, dividing it evenly, then invert onto
cheese-topped bread to form sandwiches.
Lightly oil a 15- by 10-inch shallow baking
Melt 1 tablespoon butter in a 12-inch nonstick
skillet over moderately low heat, then cook
sandwiches, turning over once, until golden, 3
to 4 minutes total. Remove from heat and
transfer sandwiches to baking pan, then wipe
out skillet with paper towels.
Top each sandwich with 1/3 cup sauce, spreading
evenly. Broil sandwiches 4 to 5 inches from
heat until sauce is bubbling and golden in
spots, 2 to 3 minutes. Turn off broiler and
transfer pan to lower third of oven to keep
Fry eggs, covered. Top each sandwich with a
fried egg and serve immediately.
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