Black Dog (2019)

Black Dog can be considered like "the daily lives of teachers", while Live was the daily lives of cops. both are taking the realistic approach to these commonly seen jobs in dramas.

both Black Dog and Just Between Lovers are similar in their slow and healing nature, both had that cold blue vibes

They are the same in the sense of vibe, and both are a story of professional worker. While Black Dog has this warm, healing feeling, Diary of Prosecutor is more quirky and portray thestruggles of adult worker life in Korea.
They air at the same period too, so I watched them hand in hand and I really liked both.

Both tells a tale about everyday life and the leads have depth and doing reflection all the time. Both are healing dramas.

in the simplest similarity Hae Won is/was a teacher kinda like Ha Neul and there seems to be a backstory, but otherwise both have that cold cozy healing vibe