Catalysis of the repair of T/G mismatches arising from deamination of 5-methylcytosine in DNA by nicking double-stranded DNA within the sequence CT(AT)GN or NT(AT)GG next to the mismatched thymidine residue. The incision is mismatch-dependent and strand-specific, in favor of the G-containing strand. The incision serves as a starting point for subsequent excision repair by DNA polymerase I, which excises thymidine and reinserts cytidine.