In a wide, shallow bowl, beat eggs and milk to combine.
Lay a slice of bread on a work surface and top with 1 slice turkey, 1 slice cheese, 1 slice ham and a second slice of bread. Repeat to assemble remaining sandwiches.
Warm a large nonstick frying pan over mediumhigh heat. Melt butter (if pan is not large enough to accommodate all 4 sandwiches, melt half of butter and cook sandwiches in 2 batches). When butter sizzles, swirl pan to coat. Carefully dip each sandwich in egg mixture, turning to lightly coat both sides, and lay sandwiches in hot pan. Cook, turning once, until sandwiches are well-browned on both sides, 4 to 6 minutes total.
To serve, cut sandwiches in half diagonally and sprinkle lightly with confectioners' sugar, if desired. Top each sandwich half with a small dollop of jam, or serve jam alongside for dipping.
