Heat a large skillet, preferably cast-iron, over medium heat until very hot. Using a paper towel, pat both sides of duck dry. Sprinkle both sides with 3/4 tsp. salt total. Lay duck in pan, skin side down. Immediately reduce heat to low.
Cook duck slowly, without turning, until a lot of fat has rendered and skin is crisp, about 20 minutes; reduce temperature if skin starts getting too dark. Meanwhile, preheat oven to 400°.
Set a cooling rack on top of a rimmed baking sheet. Transfer duck to rack, skin side up. Roast until done the way you like, about 10 minutes for medium. Remove duck from oven and let rest 10 minutes.
Meanwhile, pour duck fat into a measuring cup; if needed, add butter to make 1/4 cup. Return fat to pan, heating over low heat until it's warm. Whisk in vinegar and remaining 1/4 tsp. salt. Add tangerine segments, stirring gently to combine. Slice duck through skin into 1/2-in.-thick pieces and set on plates. Spoon tangerines with sauce over duck.
*Look for high-quality balsamic vinegar with a syrupy consistency, such as Cavalli brand (about $2 2/4-oz. bottle; amazon.com).