Hi,
 
I own a EVGA GTX 1060 6gb (quite old, but still giving me good results in gaming) which the fan just died yesterday. It was deadly noisy since a while ago, and to be honest, it has always been the weakest point about this model IMHO (noisy in-game, with a temperature around 80°C). 
 
I have an opportunity to recover a fried MSI Geforce GTX 770 Twin Frozr (PCB  and ) with both heatsink and fans in perfect condition. Though I know it would be a warranty-breaker in any possible ways  (since I own this card for a while, it does not make any difference anyway), do you think I could retro-fit the heatsink from the 770 to my 1060 ?
I am particularly concerned about cooling the VRMs and any non-in-direct-contact components, and of course, if the screws for securing the heatsink around the GPU are based on the same hole pattern in the first place ?
 
Many thanks.

Another option is a cheap parts only card off of eBay and using the fan off of one.  Which 1060 model do you have?
 
Agree, don't swap the heatsink, change the fan.

I agree that it's best to find a 1060 fan to replace it with. Odds are good that you could find videos on like youtube to show you how to replace it but I always recommend a few videos instead of just trusting one source. I also feel watching a few videos gives some tips and tricks that the other videos might not have shown to make the job easier. Or you could just find a card that is burned out but might have a working fan to buy cheaply and replace the fan on that. The only problem with going with parts from another card is that you don't know what kind of life the fan has left in it being used.