How to cook chicken cutlets in a cast-iron skillet
Chicken breasts do not require extremely high heat, nor strong searing (in fact, they are quite easy to overcook), so while a cast iron pan is certainly effective, it is not required.  Any quality pan will do.
If you are sauteing simple chicken breasts (perhaps lightly marinated or seasoned), the technique is quite simple.
Often, you are better off pan-roasting as follows:

Preheat your pan for about 5 minutes over medium high flames (or burner setting if you are electric), and oil it lightly.

Put the chicken breasts in, and leave undisturbed at medium heat until they are lightly browned on one side and released from the pan, 4-5 minutes.

Flip the breasts over and again, leave undisturbed until lightly browned.

If not done, place in a moderate (350 F, 180 C) oven until done, probably 5-10 minutes, depending on the thickness of your breasts.
You will know they are done when they register 160 - 170 F (depending on your preference; I like the lower end of the range) on a high quality instant read thermometer.   You will also learn to know when they are done based on the texture an how much resistance they have to being poked.

Once they are done, remove them from the pan and let them rest a minute or two.

During this time, you may optionally return the pan to the stove top and create a pan sauce of your choice.


While a little more work, pounding the with meat mallet (I don't own one, so I use an 8 inch skillet as a mallet) to an even thickness of no more than a little less than an inch (2 cm) will help them cook quickly and evenly.
Note:  if you have true cutlets (which are about 1/4 inch thick, less than a CM), you will need to raise the heat slightly, and only cook for 2-3 minutes.  The oven phase is unlikely to be necessary.  Otherwise, the technique is the same.