Step 1 - Prepare the Onion and Garlic: Chop the yellow onion into small, even pieces. Peel the garlic cloves, lightly smash them to loosen the skins, and mince them. Keep the onion and garlic separate because they cook at different rates.
Step 2 - Melt the Butter and Olive Oil: Place a large Dutch oven or heavy-bottomed pot over medium heat. Add the butter and olive oil, then let the butter melt completely.
Step 3 - Cook the Onion: Add the chopped onion and stir well. Cook for about 8 minutes, stirring occasionally, until the onion is soft and translucent and develops a slightly sweet flavor.
Step 4 - Add and Cook the Garlic: Add the minced garlic to the softened onion. Stir regularly and cook for about 5 minutes, until fragrant. Do not let the garlic brown or burn, as it can make the tomato bisque taste bitter.
Step 5 - Add the Tomatoes and Stock: Pour both cans of whole peeled tomatoes and all of their juices into the pot. Stir carefully, then add 3 cups of vegetable stock, chicken stock, or water and mix well.
Step 6 - Season the Tomato Bisque: Add the sugar, black pepper, and a generous pinch of salt. Add the dried thyme and dried oregano, then stir until everything is well combined.
Step 7 - Simmer the Soup: Bring the mixture to a boil over medium-high heat. Once boiling, reduce the heat to a gentle simmer and cook uncovered for about 30 minutes, allowing the liquid to reduce and the flavors to develop.
Step 8 - Blend Until Smooth: Remove the pot from the heat and use an immersion blender to blend the soup until smooth and silky. Blend only until the desired consistency is reached, as excessive blending can sometimes create a bitter flavor when olive oil is used.
Step 9 - Add the Cream: Stir heavy cream into the blended soup until it becomes rich and creamy. Add it gradually so you can adjust the richness and consistency to your preference.
Step 10 - Taste and Adjust the Seasoning: Taste the tomato bisque and add more salt and black pepper if needed. Stir well and gently reheat if necessary, avoiding a hard boil after adding the cream.
Step 11 - Garnish and Serve: Ladle the hot tomato bisque into bowls and finish with torn fresh basil. Serve warm with crunchy croutons or a piece of crusty bread.