Place a skillet over medium heat. Add 2 tablespoons of butter, and once melted add the onions. Cook, stirring frequently, until the onions are lightly golden. Remove from the skillet and set aside.
Wipe out the skillet and return it to medium-low heat.
Butter one side of each slice of bread with the remaining butter. Place two slices of bread butter side down on a cutting board. (You can assemble the grilled cheese in the skillet if you prefer. Place the bread butter side down once the skillet is hot.)
Add 2 slices of cheese to each sandwich bottom. Then top each with half of the brisket, barbecue sauce, and onions. Place another slice of cheese on top and then add the second slice of bread on each sandwich, butter side up.
Place the sandwiches in the hot pan and cover. Cook for 2-3 minutes, then flip, and cook for another 2-3 minutes until the cheese is melted and the sandwiches are golden brown.
Remove from the pan and serve warm. You may want to cut them in half first.
Notes
The brisket I used was already mixed with barbecue sauce. You can use sauced or unsauced brisket for this sandwich.