In a large bowl, toss the cabbage with the vinegar, sugar and salt. Let stand at room temperature, tossing occasionally, until softened, about 30 minutes.
Heat a panini press or griddle. Arrange the focaccia on a work surface, cut sides up. In a small bowl, combine the ketchup with the mayonnaise. Spread the dressing on the bottom halves of the focaccia. Set a slice of Gruyère on the dressing. Top with the corned beef, pickled cabbage and the remaining slices of Gruyère. Close the sandwiches and spread the outsides of the focaccia with the softened butter.
Add the sandwiches to the press and cook over moderate heat until the cheese has melted and the focaccia is crisp and golden, about 6 minutes. Cut each sandwich in half and serve right away.