1. after my friend's becoming the new manager 2. after my friend becoming the new manager 3.
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1. after my friend's becoming the new manager
2. after my friend becoming the new manager
3. after my friend became the new manager
(1) is too stilted to be usable, even if it was theoretically possible. (2) is not possible. Use (3).
(By the way, if this is from a CV, I'm not sure that it paints things in the best possible light.)