Preheat the oven to 400°F.
Slice the tomatoes in half. Transfer the tomatoes and garlic cloves to a parchment-lined baking sheet. Drizzle cooking oil and salt onto the tomatoes and garlic. Bake for 45 Minutes.
While the tomatoes and garlic are roasting, prepare the caramelized onions. Slice the onion lengthwise into long pieces. In a large pot or dutch oven over medium-low heat, add butter and a tablespoon of cooking oil, then add sliced onions. Continue to cook the onions, stirring occasionally, and checking every 5 minutes or so. If the pan begins to dry, add 1 tbsp water and lower the heat. Continue to cook the onions until they are golden brown in color and are completely caramelized, about 20 minutes.
Once the tomatoes are done roasting and the onions are caramelized, add the tomatoes, garlic, onion, and basil to a high-powered blender and blend until you reach your desired consistency. (Make sure to use a blender that can tolerate hot foods)
Pour the tomato mixture into the pot or dutch oven that was used to make the onions, add vegetable broth, and mix well. Reduce the heat to a simmer, cover, and let simmer for about 15 minutes.
Pour a generous amount into a bowl. Garnish with your favorite shredded cheese, pair with a sandwich, and enjoy!