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Preheat oven to 350°F.
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Rinse the chicken breasts in cool running water; cut off any visible excess fat, and place the chicken breasts in a baking dish.
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Season the chicken with sea salt, black pepper, and the tarragon (optional).
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Drizzle 1⁄2 tablespoon lemon juice and 1⁄2 tablespoon olive oil over each chicken breast.
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Then top each breast with 1⁄2 teaspoon dried parsley, and 1⁄2 tablespoon more of lemon juice.
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Place the chicken in the preheated oven, and bake for 35-40 minutes or until the chicken is cooked through and juices run clear (the chicken should reach an internal temperature of 165°F).
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When the chicken is nearly finished baking, prepare the Brussels sprouts.
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Wash the sprouts well, and remove and discard any outer leaves.
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Slice each sprout in half lengthwise from the stem to the top of the sprout, then brush the halves with 3 tablespoons of the olive oil, or as needed.
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Place a large sauté pan over medium heat.
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Add in the remaining 1 tablespoon olive oil, and allow it a couple of minutes to heat up just a little, then lay the Brussels sprouts in the pan, flat half facing down.
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Sprinkle the sprouts with sea salt and black pepper.
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Allow the sprouts to sauté, moving the sprouts around, so they do not stick to the pan.
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Sauté 3 to 5 minutes or until the flat side of the sprouts turns lightly golden brown and is fork-tender. (Make sure you can pierce the fork completely through the sprouts.)
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Once fork tender, turn up the heat to high, and cook for 2 to 4 minutes or just long enough for the flat sides of the sprouts to become caramelized and a deep golden brown.
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Then begin tossing/stirring and moving the sprouts around to gain a little bit of browning on the rounded sides as well.
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Transfer 1 1⁄2 cups to a serving dish, and sprinkle on a bit more sea salt and black pepper, if needed, and then finally, sprinkle the finely raw grated cheese over the sprouts.
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Serve alongside the baked chicken and enjoy!
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Let the remaining chicken and sprouts cool completely at room temperature.
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Then store in sealable containers in the refrigerator to serve as lunch the following day.
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When ready to serve, just remove from the refrigerator and reheat!