Every time I cook boneless/skinless chicken breast in a pan over the stove-top it seems to end up slightly chewy. What am I doing wrong?
Simple - you're just overcooking it. A very common fate for chicken breast.
My brother's trick as a chef is to poach the chicken breast first until just cooked (i.e. as soon as you think the breast meat is cooked through and absolutely no more).
He'll leave the breast for a few moments and the finish off in a hot pan or griddle.
Another factor may be the quality of the breast meat you're buying. It's a bit more expensive but you should always try to source free range organic chicken meat. It's way more tastier and the quality of the meat is substantially better.
I tend to pan-sear chicken in butter on medium-high to high heat until browned, then finish it in the oven at 350 degrees Fahrenheit for about 20-24 minutes. Results in much more tender, juicy chicken than fully pan cooking it.
It is to do with
(1) cooking time
(2) fat content of the chicken
(3) the way you cook it
Chicken breast contains little fat, and thus if cooked, it dries pretty easily and dry meat + heat = chewy. If you insist on the chicken breast, you can try poaching it in for example chicken stock, first, before cooking it on a pan.
You may also want to make small cuts into the chicken so that
(1) the muscle fibres are cut in short pieces which would lead to less contraction of muscle which allows for more fluid to remain in the meat
(2) more surface area for faster cooking of the chicken breast
If you do not insist on healthy cooking, using boneless chicken leg solves the problem effortlessly because of the fat.