First off, I'm going to presume that the TV is out of warranty. If it is still covered, your best bet is to call the warranty line and get it fixed that way.
It's hard to tell from your popped resistor photo, but is there a heat sink to the left of it in the image that is black? There might be something under there that blew out too.
As for DIYing the repair, you could try swapping out that through hole resistor that blew. However, while it's common for low quality capacitor to give out, resistors don't usually pop unless there is an over current situation, so there might be a larger issue at hand here. It also looks like there are 0 ohm and a 47K (? hard to read) ohm surface mount resistors in the damaged area on the back of the PCB that might need replacing.
If it's currently broken, there's no harm in attempting a repair. Find a friend who does electronics to help you source the parts, especially if you think you'll need to replace those surface mount resistors.
As for getting a replacement board, rather than having it replicated from scratch, you should be able to get a replacement board. The various boards in TVs are usually modular (power, tuner, etc), and that appears to be the power supply board. Try searching for "TVMAKE TVMODEL Replacement Power Supply Board" and see what comes up. I know the Vizio 50" plasmas have a nasty habit of blowing power supplies and thus they're pretty easy to find replacements for online.
Good luck and stay away from your boyfriend's books lest they burst into flames. ;)
Best Answer
Thermal damage. Part of chip, mostly metal, overheated, melted and evaporated, after some pressure build up it exploded. Cause is shorting. Cause of shorting can be anything, from manufacturing defect, to ESD, extraterrestrial particle, overvoltage, mechanical stress, bad design, or even software.
The chip itself is not visible on picture. What fell off is only the mold/casing material.