Cut an unpeeled orange in half. Cut one half into thin half-moons; toss with carrots, olive oil, garlic cloves, thyme sprigs, ground ginger, cayenne pepper, and 1/8 tsp. kosher salt on a baking sheet. Roast at 450°F 15 minutes, stirring once. Turn on broiler; broil 2 minutes. Discard thyme. Peel garlic; place garlic and carrots on a platter. Squeeze remaining orange half over carrots. Sprinkle with 1/4 tsp. kosher salt.