Lay two pieces of bread flat and place the cheese slices on top, keeping the cheese inside the border of the bread. Top each with a second slice of bread.
In a flat 8x8 baking dish, whisk together the eggs, milk, sugar, salt, and nutmeg.
Melt the butter in a large skillet over medium heat. One at a time, carefully set the sandwiches in the batter for a couple of seconds each, then turn once to coat the other side. Transfer to the skillet, taking care to hold the sandwiches together.
Cook the sandwiches until the cheese begins to melt and the bread turns golden, 3-4 minutes. Carefully flip the sandwiches over and use a spatula to gently press them down, concentrating on the edges to help seal them a bit. Cook until the second side is browned and the cheese is completely melted.
Transfer the sandwiches to a cutting board and cut them in half. Set on plates and dust lightly with confectioner's sugar, if desired. Serve with strawberry jam for dipping.