Preheat oven to 375°. Heat 2 tbsp. oil in a large, heavy ovenproof frying pan or wok over medium heat. Add carrots, onions, fennel, ginger, and 1/2 tsp. each salt and pepper. Cook, stirring often, until vegetables soften, about 9 minutes. Transfer to a bowl and wipe pan clean.
Boil 3 cups water. Remove from heat, add tea leaves, cover, and let steep 5 minutes. Heat chicken broth to a slow simmer in a pot. Strain tea into broth. Season broth with some salt and set aside, covered.
Dry chicken with paper towels. Rub with 1 tsp. oil. Season all over with 1/2 tsp. each salt and pepper and the coriander.
Heat same pan used for vegetables over high heat. Drizzle in remaining 2 tsp. oil, then add chicken, skin side down. Cook until skin turns very dark brown, almost black; then flip over with tongs and transfer pan to oven. Bake chicken until just opaque, in center, about 15 minutes (cut to check). Let stand until cool enough to handle.
Cook soba according to package directions. Drain and rinse. Remove chicken skin and slice meat from bones. Put 6 large, wide bowls into turned-off oven to warm. Bring broth to a simmer, covered.
Pile noodles into bowls. Top with chicken and vegetables, then ladle in broth. Sprinkle with herbs.