In a large skillet over low heat, melt 1/2 tablespoon butter. Add the onion, 1/2 teaspoon salt, and pepper to taste. Cook on low heat, stirring occasionally, until soft and golden, or about 20-25 minutes. Continue cooking, stirring occasionally, until the onions are extremely tender and the liquid is slightly syrupy, or about 8 more minutes. Remove from heat and stir in Worcestershire sauce and mustard. Transfer the onions to a small bowl and set aside.
In the same pan, add the pastrami and lightly crisp the meat until just warmed through. Remove from heat and set aside.
Lightly butter the outsides of all 4 pieces of bread. On the inside of the bread, divide half of the Cooper cheese between 2 of the bread slices. Top each piece of bread with equal amounts of hot pastrami and onions. Top with the remaining cheese, then place the remaining 2 slices of bread on top of the Cooper cheese, with buttered sides facing out.
Melt the remaining 1 tablespoon butter in a large skillet over medium heat. Add the sandwiches and cook, pressing occasionally with a spatula, until the bread is toasted and the cheese melts, or about 4 minutes per side. While cooking, add more butter to the pan if needed.