Cut orange in half. Taking one half of the orange, cut into thin half-moons. Toss with carrots, oil, garlic cloves, thyme sprigs, ground ginger, cayenne pepper, and .125 tsp salt on a baking sheet.
Roast carrots for 15 minutes, stirring once halfway through. Turn on broiler on high and broil 2 minutes. Discard thyme.
Place carrots and garlic on a serving dish and squeeze remaining orange half over carrots. Sprinkle with remaining salt.