Heat oil in a pot over medium-high heat. Add tomatoes; saute until mostly melted, 5 to 10 minutes. Add water; bring to a boil. Reduce heat to low; cook, stirring occasionally, until flavors are well blended, 5 to 10 minutes. Add eggs, stirring slowly and constantly for 1 minute. Add green onion; cook for 30 seconds. Stir in sesame oil, sugar, and salt.