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1
Preheat oven to 425F.
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2
Pat tenderloins dry and season with salt and pepper.
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3
Heat oil in a 12-inch nonstick skillet over moderately high heat until hot but not smoking, then brown tenderloins on all sides, turning with tongs, about 5 minutes total.
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4
(If the handle of your skillet is not ovenproof, wrap handle in a triple layer of foil, shiny side out.)
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5
Transfer skillet to upper third of oven and roast until a thermometer inserted diagonally into center of meat registers 155F, 12 to 15 minutes.
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6
Transfer to a platter and let stand, loosely covered with foil, 15 minutes before slicing.
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7
While meat is standing, heat butter in same skillet (handle will be hot) over moderately high heat until foam subsides.
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8
Add apple wedges and saute, turning occasionally, until tender and golden brown, 5 to 7 minutes.
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9
Transfer apples to a plate, then add chicken broth and cider to skillet.
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10
Bring to a boil over high heat and meanwhile whisk together arrowroot and water in a small bowl.
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11
Whisk arrowroot mixture into sauce and boil until thickened and reduced to about 1 cup, about 5 minutes.
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12
Remove from heat and stir in vinegar, measured salt and pepper, and any juices that have accumulated on platter.
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13
Cut meat into 1/2-inch-thick slices and serve topped with apples and sauce.
