After Kathryn Bigelow (2010), who was the second woman to win an Academy Award for Best Director?

Chloe Zhao!

In 2021 Zhao's Nomadland (2020) won the Academy Award for Best Picture and the Academy Award for Best Directing. Nomadland (2020) also won Golden Globe's Award for Best Director, the BAFTA Award for Best Direction, the BAFTA Award for Best Film, the Independent Spirit Award for Best Director, the Independent Spirit Award for Best Editing, the Critic's Choice Movie Award for Best Adapted Screenplay as well as the Critic's Choice Movie Award for Best Director.

https://en.wikipedia.org/wiki/Chlo%C3%A9_Zhao

Chloé Zhao became the second woman to win an Academy Award for Best Director when she was recognized for her work on the film "Nomadland" (2020). The film also won an Academy Award for Best Picture as well as numerous awards outside the USA. Zhao was born in China in 1982 and she has gained renown for not only directing but also writing, producing and editing many of her films.

https://en.wikipedia.org/wiki/Chlo%C3%A9_Zhao