Preheat oven to 400° F.
Melt 1 tablespoon butter in a large skillet over medium heat. Add onions, and cook until soft, but not browned, about 15 minutes.
Melt 2 tablespoons butter In a 10-inch oven proof skillet over medium heat. Remove from heat. Arrange 1/3 of the potatoes in a layer, overlapping the slices. Top with half of the onions, and half of the cheese, Sprinkle on 1/4 teaspoon salt and half of the pepper. Repeat layering with half of remaining potatoes, the rest of the onions, and the rest of the cheese. Sprinkle with remaining salt and pepper.
Top with remaining potatoes. Heat over medium heat for 15 minutes or until bottom starts to brown.
Top with piece of parchment paper, and press down to compact the layers. Bake at 400° F for 15 minutes. Remove from oven; top with piece of parchment paper, and press down to compact the layers. Return to oven for 15 more minutes.
Using a silicon spatula, loosen the edges and bottom of the Irish Pan Haggerty from the pan. Quickly flip onto a cutting board. Cut into wedges and serve.