Heat 1/4 cup olive oil in a saucepan over medium heat. Add tomatoes, sugar, and salt and cook for 5 minutes. Blend with an immersion blender until smooth.
Meanwhile, heat remaining 1/4 cup olive oil in a skillet over medium-low. Add garlic and cook until fragrant, about 1 minute. Add bell peppers, zucchini, and onion. Increase heat to medium and cook, stirring frequently, until vegetables are soft, about 15 minutes.
Pour tomato mixture into the vegetable mixture and stir until well combined. Cook, stirring occasionally, for an additional 15 minutes. Taste and season with salt, if needed.
Meanwhile, heat 1 tablespoon olive oil in a skillet over medium heat. Crack eggs into the skillet and cook until outer edges become opaque, about 1 minute. Cover, reduce heat to low, and cook until whites are completely set, about 4 minutes.
Divide vegetable stew amongst 4 plates and top each with a fried egg.