As I understand it (I am 99% I have seen this elsewhere on DIY but I can't find it at the moment), you can have multiple receptacles on a 30A 240V circuit, just as you can on a 15A or 20A 120V circuit. Assuming that's the case, the simple answer is to wire up a second receptacle and leave the EVSE plugged in all the time. However, that puts the onus on the user to make 100% certain the dryer is not running at the same time as the EV is being charged. Another answer listed some ways to have it all work automatically, at an extra equipment cost.
My take on the real issue
While I understand everyone wants to save money (myself definitely included), I think that budgeting for charging should be part of everyone's EV purchase decision.
For someone with their own home that means the cost of installing a separate circuit for charging, the cost of proper hard-wired EVSE (like the Tesla Wall Connector) and, if necessary, panel upgrade or even a service heavy-up. That cost could range from a few hundred $ to a few thousand $. Which seems like a lot, but is a tiny fraction of the cost of the EV itself. Is $1,000 too much to pay for a Tesla Wall Connector and running a circuit 30'? It is on the order of 2% of the cost of the vehicle, and is the equivalent of getting your own gas pump for an ICE, which would cost a whole lot more than $1,000!
Alternatively, for someone who lives in an apartment or other location where their own private charging station is simply not an option, that means including the cost of using Tesla Superchargers or other publicly available, but not free to use, charging equipment. Estimate the cost over the first few years of ownership and factor that into the estimated total cost of ownership.
Obviously electric vehicle manufacturers aren't going to add in this cost - it would make the cars cost more, and some people don't have any additional cost (e.g., 2nd electric car) or minimal cost. But it should be a part of everyone's purchase process.
