How would you handle an employee resigning during a critical time period?

When employees leave suddenly, it's best to let them go. Even if you have lots of work that needs to be done, having someone who has decided to move on is not the best person to have working on these projects.

If there is a team, meet with them, explain the situation and reassign work that can be done by others to get the goal met. If there is not enough people to do the extra work, bring in a consultant experienced in this area and pay them to finish the work.

Its unfortuneate that the employer is in this situation but unless the employee is leaving on good terms and is willing to finish things before leaving, you are better off, helping them out the door.

The first thing YOU should do is not to feed his panic. This is no different than someone having to leave for an emergency, or the team accepting another high profile project when the team already has a full plate. In short - this is a project management issue. And you should approach the client in a calm manner.
And - in my opinion - it's not logical to try to force the "leaving employee" who is already burned-out and has a short-term mindset train another expert and complete 2 months of work in 20 days. I don't feel that's a winning game plan.
I don't know the industry or market you are working with - so these solutions will be general in nature. Setup an appointment with me to discuss more detail solutions that are better aligned with your specific situation.

There are several project management options:
1) Reduce the scope of the deliverables to fit the current resources.
2) Reduce the complexity of the solution to fit the current resources and talent.
3) Release early and often to the end-client (your client's client). Give then end-user early drafts, demos and releases as the project progresses. Because the end-user is frequently receiving and reviewing the progress, they can tell you which features they really need by XX date, and which features they can wait on. Then schedule the features as the user needs them (versus all at once).
4) Studies show that end-users only use 36% of the entire product. Find out the features the end-user are actually going to use and postpone (reschedule the others for maintenance releases).
5) Accept the knowing that the product is never "done" - there are always going to be maintenance and improvement cycles. Make better use of the rolling delivery cycles.

Last tip is to never make someone "indispensable" - always have pair-training, pair-testing, and a buddy system. I am not recommending that everyone has to be able to do the same things at the same quality. But you should have various team members knowledgeable about other people's areas such that they can stand-in or help out.

I don't know the industry or market you are working with - so these solutions will be general in nature. Setup an appointment with me to discuss more detail solutions that are better aligned with your specific situation. You can setup an appointment at https://www.timetrade.com/book/WFSFQ

Has anyone talked to the employee to find out why he's leaving? If it's money, find out if he'd be willing to work as a freelancer for more money until the project's delivered. This gives you more time to find a permanent replacement and the project keeps moving along. Is it because the project timeline and/or workload are unreasonable? Throwing more bodies at the project may relieve the problem until the project is delivered. Is it because the project supervisor/manager is difficult or unreasonable? If it's because the departing employee doesn't want the responsibility, then you need to ask yourself why. Does your company have a culture that easily assigns blame? If so, it may be that the employee doesn't want to see his career go up in smoke because of something he has no control over.

It sounds to me like there's more going on at your client's company than one fed-up employee. For the health of your client's company, human resources should take an unbiased look the department in question and make sure it's not symptomatic of a company-wide problem.

JC- I agree with your approach above...Has your client approached the employee and asked for an extension on the resignation or possibly contract him to work with a new employee? Does the work need to be done onsite or can your client outsource to an expert?

I would start by abandoning the negative mindset. When you start with blaming an employee for "only" giving twice the customary notice in a resignation, you aren't in the right mindset to really solve the problem. As far as alternatives, consider a contract or temporary worker either to complete the project or to take on other responsibilities so the project can go more smoothly. If it's a highly technical field, there are likely contract workers available or companies that will subcontract some of the work.

There are few ways to look at it. We work in a society where people are more conditioned to leave if they get a better opportunity.
Since this employee gave a month, notice there is ample time to cross train someone to take on the additional responsibility while a replacement is found. All roles and responsibilities of the employee leaving need to be documented. A good leader will set this plan in motion.
Granted goals are important but business continuity needs to be higher priority.

I like where you are going with this JC. If the deadline is firm there is no reasonable alternative to get the task done. I'm not sure who that expert would be but I'd suggest it will be a colleague who is already reasonably up to speed on what the resigning employee is doing. If the task is not fully completed in 20 days it may be reasonable for the "expert" employee to ensure the task gets done.