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Combine the salt, sugar and water in a ziplock or reusable bag. Shake vigorously to dissolve the contents.
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Add the chicken breast(s) to the brine. [I like to make chicken breasts in bulk because they're good for adding to a salad for lunch or dinner on subsequent days.] Let the chicken stay in the brine for 45-60 minutes. I always do this at room temperature and have never had any problems, but if you live in a warm climate, you'll want to do this step in the fridge.
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Take the breasts out and dry off with paper towels. Season with lemon-pepper (or other seasoning of your choice).
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Heat the oil in a large, heavy skillet over medium-high heat. Let the oil get shimmering hot, but not smoking.
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Don an apron, and using tongs, carefully place the chicken breast(s) into the pan and cover with a splatter screen to avoid getting splatters on yourself or your stove--careful drying of the chicken breast will help with this step, but it's hard to avoid splatters altogether.
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Let the chicken breast cook (sear) at medium-high heat for about a minute and then turn and sear for another minute.
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Turn the heat down to the lowest setting and cover the pan. Let the chicken cook for abut 5 minutes. Remove the pan from heat and let the chicken sit, covered in the pan for another 5 minutes.
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This method is pretty well foolproof and will result in a juicy and tender chicken breast.
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The chicken will keep in the fridge for 3-4 days and I usually make several of them at one time and save them for future meals. They are great cold as a protein on a big salad for lunch.