Edman method to identify peptides with Phenylisothiocyanate (PTH)
The short answer is that the Edman degradation is a multi-step process. The Wikipedia page has a decent picture of the mechanism. In practice, the peptide is reacted with phenylisothiocyanate (PTH) under mildly basic conditions to give a thiourea, which is stable. The excess PTH is separated from the thiourea intermediate. The thiourea is then treated with acid to cleave the N-terminal residue from the peptide. The PTH is no longer present, so there's no concern about reacting with the second residue.