Heat the vegetable oil in a large pot over medium heat.
Add the chopped onion, minced garlic, and sliced ginger. Sauté until fragrant and the onions are translucent, about 5 minutes.
Add the chicken pieces to the pot and cook until they are no longer pink and start to brown slightly, about 5-7 minutes.
Add the chopped tomatoes and cook for an additional 3 minutes until they start to break down.
Pour in the chicken broth or water. Bring to a boil.
Reduce the heat to a simmer and cook for about 20-30 minutes, or until the chicken is tender and cooked through.
Add the green papaya or chayote slices. Continue to simmer until the vegetables are tender, about 10 minutes.
Stir in the fish sauce, soy sauce (if using), salt, and black pepper. Adjust seasoning to taste.
Add the leafy greens and cook for an additional 2-3 minutes until they are wilted and tender.
Remove from heat and ladle the soup into bowls.
Enjoy your meal!