Ask your butcher to give you chicken breasts that are all as close to the same size as possible, so they will all cook at the same rate.
Preheat your oven to 400 degrees. Place a heavy cast iron pan on the stovetop, add enough vegetable oil to lightly coat the bottom, and turn the heat on the burner to medium-high. (Cast iron pans take a few minutes to heat up, so if you are using a thinner, lighter ovensafe pan, don't heat it up until the chicken is ready to go so you can watch it and make sure the oil doesn't burn.)
Using paper towels, pat your chicken breasts dry. Season with salt and pepper on both sides. On a large plate, blend flour (start with a half-cup for two breasts, add a couple of tablespoons for each additional breast) with salt and pepper -- approximately one teaspoon salt and 1/4 teaspoon freshly cracked black pepper per breast. Dunk each chicken breast in the flour mixture, shaking to remove excess. In the photo to the left, you can see that this coating is very light.
Place chicken breasts in the pan, presentation-side down. When that side has turned golden brown, flip the breasts over, turn the burner off, and insert the probe of your thermometer into the thickest part of one of the breasts and put the pan in the oven. Set the thermometer to go off when the internal temperature reaches 162 degrees (I know, chicken should be cooked to 165 -- don't worry, even thin breasts will coast up those last three degrees while they rest.)
When the chicken has reached the desired internal temperature, remove the pan from the oven and place the breasts on a cutting board to cool. Tent with foil while they rest to keep them hot if you are planning on serving them right away. Wait at least 5 minutes before cutting the meat.
If you are allergic to wheat, cornstarch should work fine as a substitute.