Preheat oven to 400 degrees F. Brush chicken with 1 Tbsp. of the olive oil. Transfer to a foil-lined baking sheet. Finely shred 2 tsp. peel from the tangerines. Peel tangerines; slice crosswise and set aside. Sprinkle chicken with tangerine peel, 1/4 tsp. of the salt, and 1/4 tsp. of the pepper. Bake 15 to 20 minutes or until done (165 degrees F). Transfer to a cutting board; let stand 5 minutes.
Meanwhile, place cherries in a small microwave-safe bowl. Add pomegranate-blueberry juice; microwave 30 seconds. Let stand 10 minutes. Drain cherries, reserving juice; set cherries aside.
For dressing, in a small bowl whisk together reserved juice, remaining 2 Tbsp. olive oil, vinegar, remaining 1/4 tsp. salt, and remaining 1/4 tsp. pepper. Pour half of the dressing into a large bowl. Add kale and onion; toss to coat. Transfer to a serving platter.
Slice chicken; arrange over kale mixture with tangerines and pear. Top with cherries and Parmesan cheese. Drizzle with remaining dressing.