Whisk eggs, milk, vanilla, nutmeg, salt, and pepper in a shallow dish until smooth.
Stir mayonnaise and Dijon together to make a quick spread.
Spread the Dijon-mayo on one side of each bread piece.
Layer a slice of Swiss, ham, turkey, then another slice of Swiss between two bread pieces to form each sandwich.
Heat a large nonstick skillet over medium and melt the butter.
Dip each sandwich into the egg mixture, coating both sides and letting excess drip off.
Cook the sandwiches until deeply golden, 3–4 minutes per side; cover the skillet briefly to melt the cheese.
Rest 1 minute, dust lightly with powdered sugar, slice, and serve warm with raspberry jam.