Whisk minced onion, water, tangerine juice, coconut aminos, 1/3 cup coconut oil, green onions, garlic, ginger, vinegar, salt, and black pepper together in a bowl until marinade is evenly mixed. Place chicken in a separate bowl and pour 1/2 of the marinade over chicken.

Bring a large pot of lightly salted water to a boil. Add string beans and broccoli; cook uncovered until broccoli turns bright green, about 1 minute. Drain in a colander and immediately run under cold water for several minutes until cold. Drain and transfer to a large bowl.

Heat ghee in a skillet over medium heat; cook and stir mushrooms until lightly browned, 5 to 10 minutes. Transfer mushrooms to bowl with beans and broccoli, reserving ghee in skillet.

Cook and stir sliced onion in the ghee until softened, 5 to 10 minutes. Transfer onion to the bowl with mushroom mixture.

Remove chicken from marinade, reserving marinade. Heat a cast iron grill pan or nonstick skillet over medium heat; cook chicken until no longer pink in the center, about 10 minutes. Transfer chicken to mushroom mixture.

Heat 2 tablespoons coconut oil in a separate skillet over high heat; cook zucchini and carrots, using tongs to toss zucchini, until tender, 2 to 3 minutes. Add mushroom mixture, marinade from chicken, and reserved marinade to skillet; cook and stir until simmering and heated through, 5 to 10 minutes.