Macbook Pro Randomly restarting after RAM upgrade

First step in Troubleshooting to do is to start in Safe mode. (Hold Shift during start up)

That will eliminate 99% of foreign applications causing the problem.

That will also show if it is a hardware problem. Sometimes it is enough to start in Safe mode to fix small problems.

If problem continues, run the Apple Hardware test, that might show problems with hardware. (hold "D" during start up).

For more clearer picture of your system, I recommend to run the EterSoft analytical tool. EtreCheck is a simple little app to display the important details of your system configuration.

The best way to resolve hardware RAM issues is to isolate this around one of the sticks of RAM. This is most likely a hardware issue, that will only show as a bunch of random software impacts, making it really hard to diagnose.

Remove all the RAM, and put in only one new 4GB stick in, then start the Mac. If it boots and runs, then you have a good stick. Try the other one.

If the Mac won't boot with either, then try mixing a 2GB with a 4GB stick, as you already know your 2x 2GB sticks work. (sometimes you need both DIMM slots filled).