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1
Preheat the oven to 225F.
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2
Coat the chicken breast halves with a thin film of olive oil.
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3
Place each chicken breast half on a sheet of foil, cover with another sheet of foil, and pound with the bottom of a heavy skillet or a flat meat pounder until very thin (about 1/8 inch thick) and the size of a small dinner plate.
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4
Heat a large skillet (preferably cast iron) until very hot.
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5
Coat the skillet with a thin film of oil.
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6
Remove one sheet of foil from one of the pounded chicken breasts.Using the remaining foil sheet to support the chicken, lift it on one palm and flip it over (chicken side down) into the hot pan.
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7
Remove the foil.
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8
Cook just until opaque, about 30 seconds per side.
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9
When the chicken is done, transfer it to a plate and keep warm in the oven.
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10
Repeat with the remaining piece of chicken.
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11
Turn down the heat to medium-low.
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12
Coat the skillet with a thin film of olive oil and saute the shallot until transparent, about 45 seconds.
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13
Add the tarragon and the white wine.
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14
Turn up the heat to medium-high and boil until the liquid is lightly thickened, about 1 minute.
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15
Remove from the heat and swirl in the butter.
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16
Season the chicken with the pepper and drizzle the sauce over the top.
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17
Serve immediately, finishing each plate with a flourish of flake salt.
