Arrange bread slices on a work surface; spread each with about 1 tsp. jam. Top 8 bread slices with a folded half slice of cheese, the ham, then another folded half slice of cheese. Top with the remaining bread slices, jam side down.
Crack the eggs into a pie plate and add the milk. Using a fork, beat the mixture to combine. Add 4 sandwiches; let soak, turning once, until bread is saturated but not soggy, about 30 seconds per side. Transfer to a plate and repeat with the remaining sandwiches.
Melt 1 tbsp. butter in a large nonstick skillet over medium heat. Add 4 sandwiches; cook until the bottoms are golden, 2 to 3 minutes. Add 1 tbsp. butter, flip the sandwiches and cook until golden, 2 to 3 minutes more. Repeat with the remaining butter and sandwiches.
Divide sandwiches among plates and dust with confectioners' sugar. Serve with the orange and apple wedges.