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. Cannot use the alternative thiols cysteine, N-acetylcysteine, or glutathione instead of gamma-glutamylcysteine as substrates, and histidine is a poor sulfur acceptor substrate compared to hercynine. gamma-L-glutamyl-L-cysteine + hercynine + O2 = gamma-L-glutamyl-hercynylcysteine S-oxide + H2O kcat is 1.2 sec(-1). Amino-acid biosynthesis; ergothioneine biosynthesis. Monomer. Belongs to the EgtB family.