Spread the mayonnaise evenly over one side of each bread slice.
Place two of the bread slices, mayonnaise side down, on a work surface.
Top evenly with the Brie, Parmesan, arugula, lemon juice, and flaky salt.
Spread the raspberry jam evenly over the remaining two bread slices (only the sides without the mayonnaise).
Place these slices, raspberry jam side down, on top of the arugula on the assembled bread slices.
Heat a cast-iron skillet over medium-high until hot.
Transfer the sandwiches (without flipping them) to the hot skillet.
Cover and cook until toasted on the bottoms and the cheese starts to melt, about 5 minutes.
Carefully flip the sandwiches, and cook, uncovered, until toasted on the other side, 1–2 minutes.
Remove the sandwiches from the skillet, and cut each in half.
Serve immediately.
