Preheat a griddle pan or cast-iron skillet over medium heat. In a large bowl, whisk together the preserves, shallot and vinegar. Whisk in the EVOO; season with salt and pepper.
In a small saucepan, melt 3 tablespoons butter over medium heat. Whisk in the flour and cook for 1 minute. Whisk in 1 1/3 cups milk and cook until thickened enough to coat the back of a spoon, about 5 minutes. Season the sauce with salt, pepper and nutmeg. Stir in the mustard and turn off the heat.
Spread the sauce on 1 side of each bread slice. Top each of 4 slices with 1 cheese slice, 2 folded ham slices, another cheese slice and 1 of the remaining bread slices.
Beat the eggs with the remaining 2/3 cup milk and coat the sandwiches twice in the batter. Wipe the griddle with the remaining 2 tablespoons butter nestled in a paper towel, then add the sandwiches to the griddle and cook, turning once, until deep golden, about 8 minutes. Halve the sandwiches diagonally.
Add the romaine, frisee and figs to the dressing and toss. Serve with the sandwiches.