Preheat the oven to 425 degrees F. Brush the tortilla with 1 tablespoon of the butter, and sprinkle with some salt and pepper. Mix the mayonnaise and mustard together in a small bowl; set aside.
Put the hash brown on a baking sheet, spray lightly with cooking spray and bake until browned and crisp, about 15 minutes. During the last 2 minutes of baking, put the buttered tortilla on the baking sheet, and cook until warmed through and pliable. Cut the hash brown into quarters.
Meanwhile, heat a medium nonstick skillet over medium-high heat. Add the corned beef, and cook, breaking it up with a spatula, until browned and crispy, 5 to 7 minutes. Transfer it to a bowl, and set aside.
Crack the egg into a bowl, add 1/4 teaspoon salt and a grind of pepper and whisk with a fork. Melt the remaining tablespoon butter in the skillet, reduce the heat to low and add the egg. Swirl the pan so the egg forms a thin circle, and cook until just set, about 30 seconds. Roll the egg circle into a log with a spatula, and transfer it to a cutting board.
Lay the tortilla flat, and spread with the mayonnaise-mustard mixture. Arrange half the corned beef along the lower third. Top with the egg log, the remaining corned beef, the pickles, the hash browns and a squeeze of ketchup. Fold in the sides, and roll the tortilla up from the bottom.
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