Cook mortadella slice in a pan with butter until crisp on both sides and hot all way through.
Toast brioche slice in a medium-size hot pan. Slightly press brioche into hot pan until you can feel it sizzle and brown. (Brioche does not need butter to brown nicely due to its inherent butter content.)
In a small buttered cast-iron pan, crack egg and break firm inner white with a knife. Over medium-low, allow white to fully set without cooking yolk. Cut to size with a ring cutter and place on top of mortadella.
Place mortadella and egg on top of griddled brioche and plate next to smear of mustard aïoli. Garnish egg yolk with pepper, chives and salt.