I checked my Windows folder and did not see any kind of minidump file generated.
I should also mention that when I did take this machine back to the shop, he did end up replacing every part at some point with a brand new component. Same model, but a new part. He did this for the motherboard, CPU, PSU, etc.
I'm almost leaning towards a bad HDD and I'm tempted to purchase a new one and install the OS to see if that fixes the problem. The reason why I didn't think it was the HDD to begin with was that I had it in my previous machine with Windows XP and did not encounter any issues whatsoever. I believe that drive is somewhere between 8-12 months old as well.
EDIT: After updating the bios and doing a repair install of the OS, the random restarts have come back. So far have had two which happened after normal internet browsing. It really does seem as if this issue mostly happens under some kind of network load.