The best way to deal with thick, torn calluses is to not get them in the first place. I have found that when my fingers start to tear, it is best to take a good dab of aloe gel (100% pure aloe gel is best) and put it on the affected area. Let that dry (it takes a while, and then do it again.
You can GENTLY sand off (with a heel/foot pumice stone?) some of the torn area when the finger tip is completely dry, but be careful not to remove too much skin. Use the aloe to help heal the area and strengthen the skin.
Again, the best thing to do I not get one in the first place. Often, torn calluses come from thick skin that has been pulled at too hard. Practicing touch technique can help you avoid this problem to begin with AND will help speed up your left hand fingering technique.
Touch Technique is practiced like this. Instead of pressing down the strings with your left hand, just touch the string where you would normally press down. You should only touch the string hard enough to mute the note (the note should not be heard). Relax your left hand while you do this. You can practice this technique while playing fiddle melodies and scales, but be aware, if you can hear any clear note come through, you are pressing too hard.
What this will teach your hand is how to press only hard enough to get a clean tone but not so hard as to cause thick, tearing calluses. It will also help teach your left hand to relax so that you can play faster and cleaner.