Melt the butter in a large pot or Dutch oven set over medium heat.
Add the onion and stir well. Let it cook for 4 to 5 minutes or until softened.
Add the garlic, shallot, basil, oregano, tomato paste, sun-dried tomatoes, and red pepper flakes. Mix well and let it cook for another 2 to 3 minutes.
Add the onion mixture to the blender. Process until you get a smooth mixture. Pour it back into the pot.
Add the crushed tomatoes and diced tomatoes to the blender. Process until you get a smooth mixture. Pour the blended tomatoes into the pot.
Gently stir the broth into the pot and let it come to a simmer over medium heat.
Whisk the cornstarch and heavy whipping cream in a small bowl. Pour it into the pot and mix. Add in the milk and stir until thoroughly combined.
Toss the parmesan into the pot and stir until completely melted and smooth, about 6 minutes. Add the bay leaves. Reduce the heat to low and let the soup simmer for another 10 minutes.
Add salt, pepper, and sugar to taste. Stir well and remove the soup from the heat.
Ladle soup into bowls and serve warm.
Notes
Bisque: This is an elegant French name for crayfish soup, but in recent times, we have started to use the word 'bisque' to define thick and creamy soups.
Tomatoes: Because it's winter and because it's easier, I like to make this soup with canned tomatoes, including fire-roasted crushed tomatoes, fire-roasted diced tomatoes, and a few sun-dried tomatoes. If you'd like to use fresh tomatoes, I suggest roasting them and continuing with the soup recipe.