How to Make It
In a nonreactive saucepan, boil the cider and cider vinegar over high heat until reduced to 2 tablespoons, 15 minutes. Remove from the heat and stir in the butter.
Meanwhile, in a nonstick skillet, heat the olive oil. Spread the carrots in an even layer and add the garlic and rosemary. Cook over moderately high heat, without stirring, for 3 minutes. Reduce the heat to moderate and cook, stirring, until the carrots are just tender and richly caramelized, 5 minutes longer. Discard the garlic and rosemary. Season the carrots with salt and pepper. Stir in the parsley and half of the cider glaze.
In a nonstick skillet, heat the vegetable oil. Season the fish with salt and pepper and add to the skillet. Cook over moderately high heat until the fillets are lightly browned, 4 minutes. Turn and cook until the fish is just white throughout, 2 minutes longer.
Transfer the caramelized carrots to plates and set the fish on top. Drizzle with the remaining cider glaze and serve.