Melt 1 Tbsp. of the butter in a small pot over medium heat. And the onion and cook, stirring, about 10 minutes, until golden and sticky, then remove from heat and set aside. Butter two slices of bread on one side. Set butter-side down on a plate. Spread unbuttered side with the mustard and mayonnaise. Divide and place the corned beef on top. Top it with the onions. Top the onions with sliced cheese. Spread the last 2 slices of bread with butter and set on top, unbuttered side down. Heat a non-stick skillet over medium heat. When hot, set in sandwiches and cook for 3-4 minutes per side, until golden brown and the cheese is melted.