In a saucepan, melt 2 tablespoons butter over medium heat. Whisk in the flour, then 1 cup milk. Bring to a gentle boil; season with salt, pepper and nutmeg. Cook for 2 minutes to thicken. Off the heat, stir in the mustard.
Spread the bread with the sauce, then layer 4 slices with the ham, cheese and remaining bread.
In a shallow bowl, beat the eggs with the remaining 1/2 cup milk. In a nonstick skillet, melt 1 tablespoon butter over medium heat. Soak 2 sandwiches in the egg mixture. Add the sammies to the skillet and cook, turning once, until golden, about 5 minutes. Repeat with the remaining butter and sammies.
In a bowl, toss the arugula, lemon juice and EVOO; season with salt and pepper. Serve with the sammies.