M3mn0ch_1987 suggested this update to the post to make the method more clear:
Before we start: Your car still needs to be showing as destroyed in your list of vehicles when you call your mechanic.
So the steps are as follows.
1. Regardless of if you are already outside of your garage in a car go back into your garage and select a car to destroy (this would be best if it was the cheapest one that you have but I wouldn't recommend a bike as these can be delivered to you too fast to complete the glitch)
2. Take this car far enough away from your garage to be eligible for a delivery from your mechanic. (I live in the 205k Apartment so I do this at the life invader building). Destroy the car (You can kill yourself at the same time so as not to attain a wanted level / call Lester to get rid of the wanted level).
3. Go to the closest wall and face it so your back is to the road. Go to your phone book and call Mors Mutual. As soon as you press X / A to confirm that you want to call them continue to spam X / A until the call has ended (try spamming it as fast as possible)
4. As soon as Mors have hung up press and keep hold of Select / Back until the quick action menu pops up and scroll up / down until you get to Request Personal Vehicle.
5. *IMPORTANT* Only press request personal vehicle once and leave the quick select menu open until the car has arrived.
Please note - You do NOT need to call your mechanic throughout this whole procedure. I have a feeling that people are calling their mechanic rather than using the request personal vehicle option in the quick menu.
The car that gets delivered should be a COPY of the car that is being listed as "Destroyed" in your mechanic's list. If this fails (which it has always worked first time for me) then you will need to repeat the process until you are successful.
Please note - The copy car that gets delivered to you will have NO insurance so either get your backside to LSC as quickly as possible or do this in a invite only lobby.
The original post is below.