If you have signed non-compete clauses in the employment contract, then you may be prevented, for the specified period, from contributing ideas related to your work to any project whatsoever anywhere, open source or proprietary. That has nothing to do with the particulars of that project.

Just because the employers approved some code to be released under the MIT, which allows for unfettered competition using the ideas embodied in that code, doesn't mean they approve the leakage of other ideas (either into that code, or any other code).

Nothing in the MIT license itself prevents you from forking the project; from that perspective, it's no different from any other individual's or organization's MIT-licensed program. It's all about what you are allowed to do according to the entirely separate contract with the employer.