In a small bowl, whisk together the egg and milk. Dip the bottom side of each sandwich into the egg mixture, then transfer to a hot grill pan. Dip the remaining two pieces of bread into the egg mixture, then place egg-side up on top of the sandwich. Cover the grill pan and allow to cook over medium heat until the brie begins to melt and the bottom bread is golden brown. Flip to the other side, cover, and cook until bottom is golden brown.