It seems to me that becoming good at mixing and mastering is mostly the process of your ears becoming more sensitive to small changes in sound. When I first opened up an EQ, I couldn't hear any difference to anything but volume.
I'm a bit better now, but I gave up trying to watch a 3 hour mastering class the other day because I couldn't hear the difference between the before and after tracks.
Is there anyway to speed up this process at all? How do the pros get such sensitive ears?
time is the only path you can walk to improve your ears along with good coordinated training, most professional musicians who studied music at college/university are already familiar with this paradigm as they would have undergone serious ear training to be able to recognise melodic, rhythmic and harmonic ideas in context - macgamut and teoria.com offer excellent programs of training for this type of listening.
for the more general skill of hearing audio and identifying areas that need addressing by frequency, you can buy into the paid for programs and apps or develop your own system to begin your training before embarking on a program of (paid) validation/verification
you can, of course, develop your own regime where you start with gross examples - say 12dB boost at 16kHz, 8kHz, 4kHz, 2kHz, 1kHz, 500Hz, 250Hz, 125Hz, 63Hz or 12dB cut at the same freqs and gradually shifting the amount of cut/boost down to +/- 1dB, probably through the pathway of 9dB, 6dB, 5dB, 4dB, 3dB, 2dB, 1dB. You should also be using multiple center frequencies and vary applying shelving or peaking and different bandwidth/Q factors as you go.
In all cases you should be applying this to music that you know intimately to ensure that you can actually hear the changes being made once you get down to the small values. You would probably need a dozen or so reference tracks from a range of styles/genres that you are familiar with and are likely to be working on in your professional practice. it should go without saying that these reference tracks should be 16/44.1 wav or better.
Once you get really good at it you should be able to benefit from these on-line apps directly
Another thing to remember about most mastering engineers (given that you mention mastering) is that they have a wide experience of music both as engineers and as musicians in orchestras or bands, i.e. their ears are already trained to hear music in context and they are able to pick out flaws in their own playing and others as well.
it is probable that the 10000 hour rule applies here as well - a week to develop the skill, but 10000 hours to refine and make it second nature