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1
Adjust oven rack to middle position and heat oven to 375 degrees. Whisk 1 1/2 cups apple cider, corn syrup, maple syrup, mustard, vinegar, pepper flakes, 1/8 teaspoon salt, and 1/8 teaspoon pepper together in medium bowl. Place flour in pie plate, then season chicken on both sides with salt and pepper. Working with one chicken breast at a time, coat chicken with flour, patting off excess.
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2
Heat oil in ovenproof 12-inch skillet over medium heat until shimmering. Add chicken breasts skin-side down; cook until well browned and most of fat has rendered from skin, 8 to 14 minutes. (If after 3 minutes you don't hear definite sizzling, increase heat to medium-high. If after 6 minutes chicken is darker than lightly browned, reduce heat slightly.) Turn chicken and lightly brown other side, about 5 minutes longer. Transfer chicken to plate.
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3
Pour off all but 1 teaspoon fat from pan. Add shallot and cook until softened, 1 to 2 minutes. Increase heat to high and add orange juice mixture. Simmer, stirring occasionally, until syrupy and reduced to 1 cup (heatproof spatula should leave slight trail when dragged through glaze), 6 to 10 minutes. Remove skillet from heat and tilt to one side so glaze pools in corner of pan. Using tongs, roll each chicken breast in pooled glaze to coat evenly and place skin-side down in skillet.
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4
Transfer skillet to oven and bake chicken until thickest part of breasts registers 160 degrees on instant-read thermometer, 25 to 30 minutes, turning chicken skin-side up halfway through cooking. Transfer chicken to platter and let rest 5 minutes. Return skillet to high heat (be careful-handle will be very hot) and cook glaze, stirring constantly, until thick and syrupy (heatproof spatula should leave wide trail when dragged through glaze), about 1 minute. Remove pan from heat and whisk in remaining 2 tablespoons apple cider. Spoon 1 teaspoon glaze over each breast and serve, passing remaining glaze at table.
