Heat a cast iron or stainless steel pan over high heat. Sear the lemon slices and two lemon halves, cut side down, in the pan until browned, about 3 mins.
(flip the slices once).
In a cocktail shaker, add the egg white, Calvados, ginger liqueur, bitters, and 1 1/2 oz of the charred lemon juice and shake vigorously for 30 seconds.