I recommend Kim Eye Clinic in Yeongdeungpo, Seoul.

I was treated at a small local hospital before because of a sty.

I was once shocked when I was told I needed to go to a large hospital for a biopsy. After asking around, I found Yeongdeungpo Kim Eye Clinic. It's incredibly crowded, both on weekdays and especially on weekends, so you absolutely must make an appointment in advance. This made me realize that it's a place I can trust and receive treatment.