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. gamma-L-glutamyl-L-cysteine + hercynine + O2 = gamma-L-glutamyl-hercynylcysteine S-oxide + H2O kcat is 0.87 sec(-1) for the oxidative sulfurization of hercynine with gamma-L-glutamyl-L-cysteine as sulfur donor. kcat is 0.25 sec(-1) for the oxidative sulfurization of hercynine with N-glutaryl-cysteine as sulfur donor. Amino-acid biosynthesis; ergothioneine biosynthesis. Monomer. Belongs to the EgtB family.