Core and slice the apple; toss in 1 tablespoon of the lime juice. Place carrots and cabbage in a large bowl and cover with boiling water. Let sit 5 minutes; drain and cool slightly.
Whisk together oil, shallot, the remaining lime juice, salt, and cumin; add freshly ground pepper to taste.
Toss carrots, cabbage, and apple in a large bowl with dressing, mint, and sunflower seeds. Serve at room temperature or chilled.