Heat 2 tablespoons olive oil in a large skillet over medium to medium-high heat. Add tomatoes, bell pepper, carrot, and garlic cloves. Cook and stir until softened, about 15 minutes. Stir in sugar.
Fill blender or food processor halfway with the tomato mixture. Cover and hold lid down with a potholder. Blend until smooth, adding 1/2 the basil and pouring in 1/2 the vegetable stock slowly. Transfer soup mixture to a large pot. Repeat with remaining tomato mixture, basil, and vegetable stock.
Bring soup to a simmer over medium heat. Continue simmering until flavors combine, about 20 minutes. Add salt, pepper, and remaining olive oil.