Melt 1 teaspoon of the butter in a skillet over medium-low heat. Sauté the pastrami for 2 minutes per side. Lay the pastrami on the bottom half of the English muffin.
Melt the remaining butter in the skillet over medium heat until it is foaming. Add the egg, breaking the yolk with a fork, and fry, turning once until crisp around the edges, about 4 minutes; the yolk should still be runny. In the last minute, lay the cheese over the egg so that it melts. Season with salt and pepper.
Slide the egg and cheese over the pastrami, and cover with the muffin top. Enjoy immediately.