I'd like to record audio coming out of a DJ mixer that has only one output. So I thought I'd use those 'Y' shaped cable splitter plugs which take the L and R cables as input and output to two. Assuming I use decent quality splitters and cables, will this give me broadcast-quality audio?
Not in principle. Assuming the input impedance of both recipients is substantially smaller than the output impedance of the mixer (which it normally is), both do ideally recieve exactly the same signal that each of them would if connected alone.
However in practise, there is one problem which turns up really often if you have audio inputs of different devices in parallel: interference introduced inductively through a ground loop. This is usually either mains hum or artifacts from a switched-mode power supply; both can be pretty nasty. Balanced connections are much less likely to suffer from this, so if the mixer's outputs and both(!) devices' inputs are balanced it should be fine; of course only if you use a fully symmetric splitter. (In the case of TRS connection this means all of the plugs on both Y-cables need to be stereo.)
If any of the devices does not support balanced and you do experience an untolarable amount of static, the canonical solution is a DI-box. I'd recommend the ART DTI.