Place a wok or large nonstick skillet over medium-high heat.
Add 1 teaspoon oil plus spinach and cook until spinach wilts. Remove spinach to plate. Set aside.
Add 1 1/2 teaspoons oil to wok. Stir in onion and cook for 2 minutes. Stir in half the salt. Add ginger and cook for 45 seconds.
Sprinkle chicken with remaining salt and add to wok with remaining oil. Cook until chicken is opaque, 2 to 3 minutes. Reduce heat to low.
In a small bowl, whisk together broth, soy sauce, and mirin; add to wok. Stir in spinach and cook until mixture begins to bubble gently. Pour eggs over top of chicken & spinach mixture; cover and cook until eggs set, 2 to 3 minutes.
To serve, spoon a half cup of rice into each of four deep soup bowls. Top with chicken-egg mixture and serve with steamed veggies.