In a skillet over medium heat, heat oil. Add chicken liver in a single layer and cook until lightly browned. Turn and continue to cook until lightly browned but not cooked through. Remove from heat and set aside.
Remove excess oil from pan except for about 1 tablespoon. Add onions and garlic and cook until limp.
Add gizzard and cook, stirring regularly, until lightly browned.
Add fish sauce and cook for about 1 to 2 minutes.
Add water and bring to a boil, skimming scum that floats on top.
Lower heat, cover, and cook for about 25 to 30 minutes or until gizzard is fork-tender. Add more water in ½ cup increments as needed to maintain 1 cup of liquid.
Add carrots and cook until half done.
Add green peas, quail eggs, liver, and cream, stirring gently to combine.
Continue to cook for about 4 to 6 minutes or until vegetables are tender yet crisp, eggs are heated through, and sauce is slightly thickened.
Season with salt to taste. Serve hot.