Whisk the eggs with a pinch each of salt and pepper in a small bowl. Heat the peanut oil in a wok or large saucepan over high heat. Add the eggs and cook, undisturbed, until bubbles form on the surface and the bottom is set, about 1 minute.
Add the tomatoes and cook, stirring gently to break up the eggs, until the tomatoes begin to soften, 2 to 3 minutes. Add the chicken broth, 3 cups water, the scallions, sesame oil, soy sauce and 1 1/2 teaspoons salt. Partially cover and simmer 5 minutes.
Stir in the chicken, spinach and rice and warm through. Season the soup with salt and pepper.