Any recommendations for a drama where the FL is really good at her job and the ML chases her first, he’s her subordinate, The ML have a cold personality just like a butler/secretary.

Mother of Mine - has such a story. She is the middle daughter,  very successful and meets someone who is new in the firm. he works under her, but love is on the horizon. its a very well done drama :) 

She is not good in her job but everything else is accomplished https://mydramalist.com/78955-love-beyond-time-and-space

 JDramaSaikoTWT:

Any recommendations for a drama where the FL is really good at her job and the ML chases her first, he’s her subordinate, The ML have a cold personality just like a butler/secretary.

Into the Ring - Whether she's "really good" at her job is maybe debatable, but I think you could argue that she is, and she's definitely passionate. I wouldn't describe the ML as "cold" but he's quiet and likes the FL first. He becomes her secretary for part of the drama. 

Witch's Romance - The FL is a workaholic and good at her job. She suffered an emotional blow several years prior and became cold / jaded / threw herself into her work. The ML also previously suffered an emotional blow that prompted him to give up his initial career path and to start an errand running business with a friend. He is generally warm and kind, but underneath he is still hurting. Through the errand running business, the ML ends up getting hired to work under the FL for a period of time. He chases her first. 

Bloody Romance - IMO the FL was terrible at her job, but she still managed to be successful and in the end she survived, which was basically winning given the circumstances (but then again the ML had a horrible ending and it seemed like the FL was a bit crazy in the end, so not much of a win). The ML was not warm, but he also wasn't super cold. He was very quiet, and he was good at his job prior to meeting the FL and then as her assigned "shadow", which was a sort of cross between a bodyguard, an assistant and a mentor to the FL at her command.

Hotel del Luna - The FL is the ML's boss. She has become vain / seems like she doesn't care about certain aspects of her job, but she's powerful and capable when it comes to dealing with certain problems. The ML is not cold, but he also isn't excessively energetic or clingy. 

The Legends - The ML is quiet but never cold to the FL when he knows it's her. He's fairly cold to other women. The FL becomes a leader and the ML is her very junior subordinate for a while. Their roles sort of reverse later on, but when the ML knows the FL is the FL (there's a time where he doesn't, with good reason) he always defers to her. 

Search: WWW - The FL is a senior executive at a large company and is very good at her job. The ML isn't her direct subordinate, but is the CEO of his own small firm which is hired by the FL's company. The ML isn't cold, but has a quiet personality. He chases the FL first. 

The Beauty Inside - The FL is a top actress. The ML is an executive at a large company (an airline iirc) that hires the FL as its spokesperson. In their business partnership, there isn't a super clear senior and subordinate and they each have their own secrets / leverage over each other. The ML is cold and falls for the FL first (iirc he chases her first too). 

Healer - Overall the FL isn't particularly amazing at her job, but she has some good moments. The ML's main profession is as an anonymous man for hire (sort of like a reverse hitman, the only job he doesn't take is killing) with the code name Healer. In that persona he's a loner and fairly cold. He is hired to watch over the FL and ends up going under cover as her hapless, scaredy cat new assistant. Although the FL technically had a crush on the ML's Healer persona first, I'd still say the ML falls for and chases the FL first. 

Across the Ocean to See You - The FL is good at her job and becomes the ML's boss and new neighbour. When he finds out she's pregnant and single (recently divorced), he basically decides to become her personal assistant to help her with the pregnancy