Place the shredded cooked beef brisket in a small mixing bowl and drizzle it with the barbecue sauce. Sprinkle the kosher salt and freshly ground black pepper over the brisket, then toss gently with a fork until the meat is evenly coated and lightly moistened.

In a separate small bowl, stir together the mayonnaise and Dijon mustard until you have a smooth, creamy spread. Set this aside to use as the sandwich sauce.

Lay the sourdough bread slices out on a clean work surface. Spread the softened unsalted butter evenly over one side of each slice, making sure to reach all the way to the edges for even browning.

Flip all of the bread slices so the buttered sides are facing down. On four of the slices, spread a thin, even layer of the mayonnaise-Dijon mixture over the unbuttered sides.

Place one slice of sharp cheddar cheese on top of the mayonnaise-coated side of each prepared bread slice, followed by one slice of provolone cheese on each. This creates a base layer of cheese that will melt around the brisket.

Divide the seasoned brisket evenly among the four prepared bread slices, piling it gently on top of the cheese. Sprinkle the thinly sliced red onion over the brisket, then scatter the chopped fresh parsley evenly over the onions.

Top each sandwich with the remaining four slices of sourdough bread, placing them buttered-side up to form four complete sandwiches. Press down very gently with your hand to help the layers settle without squeezing out the filling.

Heat a large nonstick or cast-iron skillet over medium-low heat. Add the olive oil to the skillet and swirl or brush it around so the bottom of the pan is lightly coated.

Place two of the sandwiches in the hot skillet, buttered-side down. Cook for 3 to 4 minutes, or until the bottom is deep golden brown and crisp and the cheese is beginning to melt. Adjust the heat as needed to prevent scorching.

Carefully flip each sandwich with a spatula and cook for another 3 to 4 minutes on the second side, until the bread is evenly browned and the cheese inside is fully melted. Transfer the finished sandwiches to a cutting board and let them rest for 1 to 2 minutes.

Repeat the cooking process with the remaining two sandwiches, adding a small drizzle of olive oil to the skillet if it looks dry. Again, cook until both sides are golden and the cheese is melted, then let the sandwiches rest briefly on the cutting board.

Using a sharp knife, slice each sandwich in half on the diagonal to reveal the melted cheese and brisket layers. Serve the brisket grilled cheese sandwiches warm, while the cheese is still gooey and the bread is crisp.