Whisk sugar, salt, vanilla, milk, and egg until smooth. Add flour and baking powder and stir just until a smooth batter forms.
Preheat a panini press and lightly butter the plates. Pour a thin layer of batter onto the bottom plate. Add a generous layer of filling, then cover with a small amount of additional batter to enclose.
Close the press and cook until golden and set, 3 to 5 minutes. Open carefully and transfer to a plate. Repeat with remaining batter and filling. Serve immediately.