These sandwiches are a whimsical take on the classic Jewish deli sandwich, made Irish with the addition of a warm potato salad. Slideshow: More Irish Dinner Recipes
In a medium saucepan, cover the potatoes with salted water and bring to a boil. Boil until the potatoes are tender, about 20 minutes.
Meanwhile, in a bowl, stir together the cabbage, mayonnaise, ketchup, vinegar, shallot and dill and season to taste with salt and pepper.
When the potatoes are cool enough to handle, peel them and cut into chunks. Toss the potatoes with the cabbage salad in the bowl.
Make 2 sandwiches with the bread, corned beef, potato-cabbage salad, and cheese.
In a large heavy skillet, heat the butter over medium low heat until hot. Add the sandwiches, pressing down and turning once, and cook, covered, until the bread is golden and the cheese is melted, about 6 minutes total. Serve.