The “and I must find her” here is the tricky part. There are lots of folks with missing parents, but many fewer actively searching for them.

But, that said, I’m going to put Healer front and center here as the absolute best recommendation I have for this theme. I can’t really say a whole lot more for fear of spoilers, but I think this is a more-than-solid choice for our question-asker! I mean, it’s a solid choice for anytime you want to (re)watch a drama at all!!

Missing the Other Side : It's kinda the theme of the drama but the reunions are as sad as happy. Surprisly, I liked both seasons.

Her Private Life had Ryan Gold looking for his mom.
My Daughter, Guem Sa-wol.
King of Baking, Kim Tak-gu.
For long lost father, I'm hesitantly naming Monthly Magazine Home.

And many more.

Tale of the Nine-Tailed - FL was looking for her parents.

City Hunter had a child stolen from his mom. The mom wasn't exactly missing but they were separated for a bulk of the show.

Tunnel also had a child separated from her parents but with time-travel added to the mix of trying to bring the family back together.

Fight My Way had a non-main character searching for his mom.

Vincenzo is one where he looks for his mom after he comes back to Korea. Not a huge part of the story, but there are some great moments when you’re not sure who knows about which relationships.

Lawless Lawyer is one where the FL has a missing mom. I don’t know if you could say she’s really looking for her, but you do get a lot of the same elements in that part of the story. Some folks are not into this drama, but if you like campy action and Lee Joon-gi it’s worth a watch! I found it hilarious and great.

Thirtynine is one where you have a person who grew up in an orphanage looking for her biological mom. Not my favorite drama, but certainly not deserving of as much hate as it got, and there are some nice storylines here.

Oh! I just remembered Kairos (I think that's the title)

The mom isn't long lost as in missing for years but she does randomly go missing and the female lead's main focus is searching for her mother.

I was thinking of King of Baking, which @jerrykuvira already suggested. The missing mother/father/son was the main arc of the drama.

Other dramas I saw which contain that sub-plot are: Queen Seon Deok, Shining inheritance (missing brother and father), Six Flying Dragons.

I don't remember much about dramas that has this trope. Switched at birth, sure. Just search for old dramas. There's a bunch of them. 49 Days and Agency recently both lost their mother.

Autumn in My Heart 😅

Are You Human Too

Love all, play: female lead stalks her mum at one point.
Healer definitely qualifies here.
Mother of mine.
Crash course in romance.

The mom quest is central to Arang and the Magistrate.

Mother also had a lost and found (but not searched for) mom.

Okay so... Mystic Pop Up Bar sort of counts... But... It's not this exactly??? But it is? But [spoilers]. But you should still watch it if you like this trope and these kinds of themes, yis.

Fanletter, Please: Little Yuna, with no mother, writes fanletters to popstar KangHee

Is there some aspect of finding mom in "Good Doctor"?

Legend of the Blue Sea! The long-lost mom trope was excellent in this drama, it didn't even feel like a trope. It was heartfelt and artfully done. It moved me so much that it's the main reason the drama is on my all-time favorites list.

What a coincidence! I just finished watching My Perfect Stranger literally 15 minutes ago. It has a long-lost mom, plus some other mind-bending family reunions, all given a unique twist by a time-travel plot. Without giving too much away, I can say that the character in question has definitely not been searching for his mom.

BTW - Really loved the show!

Abyss had that subplot but with the second female lead.
The Veil, Fl had a missing father and got close to ML hoping it can give her some help find him.

Mother is the most obvious one for me. I mean – the title of the drama is “Mother” – it doesn’t get more direct than that right? I must say that when the character and the viewer gets hit with the revelation of who Soo-Jin’s (played by the ever glamorous Lee Bo-Young) real mother is…..WOW! That scene? The use of the bike lock and the key. This was probably one of the most shocking WTH moments on any show. At the moment of realization, when the key fit and the lock unlocked and the realization hit both the character and the viewer – I was screaming at the TV. It was the emotional equivalent of getting hit by the white truck of doom.
On another note – a bit sad that that show was recently removed from Viki. I was going to rewatch that scene 8(.

Dae Jang Geum! Though not exactly a lost mother, but a woman whose loss of her parents early in life drives her to pursue the excellence practiced by her parents in everything she does. I started rewatching this because despite my hatred for cooking and having both parents, I can get behind anyone with noble intentions to excel in their purpose in life, whatever that may be. Plus LYA is a brilliant actress, whom I'm eagerly awaiting her upcoming drama.

The drama that immediately came to my mind is LOVE IN THE MOONLIGHT aka MOONLIGHT DRAWN BY CLOUDS. The female lead is searching for her long-lost mom. It is one of my favorite dramas because it has it all: humor, romance, angst, intrigue, and Park Bo Gum! The OST is really great too.