When cooking chicken bought in pieces (quarters), what is the best way to achieve a dish that is without the bones?
Should you pull the meat off the bones before or after cooking?
It depends what you are making and how you are cooking the chicken:
If what you intend to create is some sort of long-cooked "pieces of meat in sauce" then leave the meat on the bones and simmer for an hour or so. The bones will give much more flavour. When the meat is tender, remove it from the bones, which will be easy, cut it up and put it back in the sauce, and keep going. I do this for curry, stew, and the like.
If you intend to give each person a recognizable piece of chicken, such as a grilled breast or a grilled leg, leave the bones in, serve it on the bone, and let the person deal with it.
If your plan is to cook the chicken for a very short time, such as a stir fry, then remove it from the bones first and cut it up. It will have less flavour but it will cook quickly.
Chicken breast is easiest to debone, and legs are harder - I use legs for the long-cooking and leave the bones in, and breasts for the short cooking and debone and cut them up first.
Either way should be fine. But consider this: If you are going to debone a chicken, why would you buy it in pieces? If you buy a whole chicken you'll get more eats for each € or $ by deboning it yourself.
So, if you bought pieces, just cook and then take the meat off, and next time buy a whole chicken instead.