How to Make It
Score a 3/4-inch diamond pattern in the skin of each breast half with a sharp knife (cutting through the skin, but not into the flesh). Sprinkle both sides of breast halves with salt; let stand 15 minutes.
Combine juice, stock, and vinegar in a medium bowl. Combine sugar and water in a medium saucepan over medium-high heat. Bring to a boil, stisrring until sugar dissolves; cook, without stirring, until liquid turns a golden amber color (about 5 minutes). Carefully stir in juice mixture (the liquid will boil rapidly); stir until hardened sugar dissolves. Bring to a boil; cook until reduced to 3/4 cup (about 10 minutes). Remove from heat; stir in rind.
Place duck breasts, skin side down, in a large skillet. Place pan over medium heat; cook 10 minutes or until skin is golden brown, occasionally pouring off excess fat. Flip breasts; cook 4 minutes for medium (145°) or until desired degree of doneness. Place breasts on a cutting board.
Pour off fat from pan. Place pan over medium-high heat. Add shallots; cook 45 seconds, stirring frequently. Add liqueur; cook 15 seconds or until liquid nearly evaporates, scraping pan to loosen browned bits. Stir in juice mixture. Bring to a boil; cook 2 minutes or until sauce thickens slightly. Remove pan from heat; add butter, stirring until butter melts.
Cut duck breasts across the grain into 1/2-inch-thick slices. Serve with sauce.