For the batter: In a shallow baking dish or pie plate, whisk flour, baking powder, salt, and nutmeg until combined. In a small bowl, whisk milk and egg until combined. Whisk milk mixture into the flour mixture until a smooth batter is formed. Batter should be thick enough to coat the back of a spoon; set aside until ready to use.

For the Sandwiches: Place 4 slices of bread on a clean work surface. Spread 2 slices with mayonnaise and 2 slices with mustard. Set aside.

Layer ¼ cup (about 1 ounce) of the cheese on each mayonnaise-coated bread slice, top with the ham, divided evenly between the 2 slices. Top with the remaining cheese (¼ cup per sandwich). Close each sandwich with the remaining bread slices, mustard side down. Press firmly to adhere.

Working with 1 sandwich at a time, use a fish spatula and hands to dip and turn the sandwich in the prepared batter to coat evenly. Transfer the sandwich to a large plate and repeat with remaining sandwich.

In a large nonstick skillet, melt butter over medium heat, swirling to coat the pan evenly, until foaming. Carefully transfer the sandwiches to the skillet and cook until golden brown on bottom, 3 to 4 minutes. Using 2 spatulas, carefully flip the sandwiches. Set a cast-iron skillet or Dutch oven directly on the sandwiches to press them. Continue to cook until the bottom side is golden brown, 3 to 4 minutes.

Transfer sandwiches to a clean wire rack set in a rimmed baking sheet and let cool for 5 minutes. Cut sandwiches in half and dust with powdered sugar, if using. Serve with raspberry jam on the side, if using.