How to Make It
Top half of the bread slices with the Gruyère and ham. Trim the overhang and add to the sandwiches. Top with the arugula and Cheddar.
In a bowl, beat the eggs with the water. Dip 1 side of the remaining bread slices into the egg; top the sandwiches with the bread, egg side up, and sprinkle with the parsley, tarragon and thyme.
Melt 1 tablespoon of the butter on a cast-iron griddle. Add the sandwiches, egg side up, and cook over moderately low heat until golden on the bottom, about 4 minutes; transfer to a work surface. Melt the remaining 1 tablespoon of butter on the griddle. Return the sandwiches to the griddle, egg side down. Cook, pressing lightly with a spatula, until golden and the cheese is melted, 3 to 4 minutes longer. Cut the sandwiches in half and serve with mustard.