Heat vegetable oil in a soup pot over medium heat; cook and stir cucumbers until softened, about 5 minutes. Cook and stir green onions with cucumbers until green onions are softened, about 5 more minutes. Pour in chicken broth and lemon juice. Stir in sugar and bring soup to a boil.
Reduce heat to low, cover soup, and simmer until vegetables are tender, about 20 minutes. Season with salt and black pepper.