The mobile "charger" as others have indicated is not a charger, it's an EVSE. Or in other words, a glorified extension cord. It seems to be exactly what you are asking for. With the "common" connector being the Tesla plug on the car
The EVSE is actually a smart extension cord. It communicates with the car and only applies power when safe to do so.
The Tesla mobile adapter (or wall adapter) is capable of [email protected]
That's a LOT more than your Home Depot extension cord is capable of. The closest comparison is the NEMA 14-50 cord commonly used in travel trailers. And they aren't cheap either.
So indeed the mobile adapter is exactly what you are looking for.