Catalyzes the oxidative sulfurization of hercynine (N-alpha,N-alpha,N-alpha-trimethyl-L-histidine) into hercynyl-gamma-L-glutamyl-L-cysteine sulfoxide, a step in the biosynthesis pathway of ergothioneine. Ergothioneine is an antioxidant that protects mycobacteria from oxidative stress. gamma-L-glutamyl-L-cysteine + hercynine + O2 = gamma-L-glutamyl-hercynylcysteine S-oxide + H2O Amino-acid biosynthesis; ergothioneine biosynthesis. Monomer. Belongs to the EgtB family.