Use paper-thin sliced prosciutto to wrap the duck, and sub chicken thighs for wild duck if needed.
Combine first 3 ingredients in a zip-top plastic bag; add duck and seal. Cover and chill 8 hours.
Preheat oven to 400°. Remove duck; discard marinade. Pat duck dry; sprinkle with Cajun seasoning. Wrap each duck breast with 1 prosciutto slice.
Brown duck in hot oil in a skillet over medium-high heat 2 minutes on each side. Place duck on a wire rack in a jelly-roll pan.
Bake at 400° for 6 minutes (10 for chicken) or until a meat thermometer registers 145° or to desired degree of doneness (165° for chicken). Let stand 10 minutes. Serve with rice and gravy.