Remove and discard giblets and neck from turkey, or reserve for another use. Trim excess fat. Tie ends of legs together with twine. Starting at neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat. Combine 1/4 cup butter, shallots, paprika, garlic, 3/4 teaspoon salt, and 1/4 teaspoon freshly ground black pepper. Rub butter mixture under loosened skin over breast and drumsticks. Lift wing tips up and over back; tuck under turkey. Place turkey, breast side up, in a roasting pan. Bake at 450° for 30 minutes.
Reduce oven temperature to 350° (do not remove turkey from oven). Bake turkey at 350° for 1 1/2 hours or until a thermometer inserted into meaty part of thigh registers 165°. Remove turkey from oven; let stand 30 minutes. Remove and discard skin.
Combine carrot, broth, vinegar, and honey in a large skillet; bring to a boil. Reduce heat, and simmer 55 minutes or until liquid evaporates and carrot is tender. Remove from heat; stir in chopped chives, 1 tablespoon butter, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon black pepper.