So, if you start holding left-click with a grenade out then immediately right-click, the you will hear the click of the lever before the pulling of the pin, and the timer starts before the pin is pulled.

Another interesting consequence, which I ran into is that if you stutter with whichever mouse button you are using to pull the pin, you can actually release the lever and pull the pin with the same mouse button without ever throwing it.

In my case, I had an incendiary grenade that I wanted to toss. I right-clicked twice accidentally instead of once. Although I did hear the noise indicating the grenade was live, I did not understand that was the case, because I had never left-clicked, which is the typical method of releasing the lever when tossing.