Melt the butter with the olive oil in a heavy-bottomed pot over medium heat. Add the onion and grated carrot with a pinch of salt and cook 6 to 8 minutes, until soft and translucent. Don't let them brown.
Stir in the garlic and cook 45 seconds, until fragrant.
Add the tomato paste and stir constantly for 2 minutes, until it darkens from bright red to brick. This step builds the savory depth.
Add the crushed tomatoes, broth, sugar, salt, black pepper, and red pepper flakes, scraping the bottom of the pot. Bring to a gentle simmer and cook uncovered 20 minutes.
Turn off the heat and rest 5 minutes. Blend with an immersion blender for a full 60 to 90 seconds, until completely smooth with no texture remaining.
With the burner off, stir in the heavy cream, sliced basil, and Parmesan if using. Taste and adjust salt, adding another ½ teaspoon of sugar if it reads sharp.
Ladle into bowls and top with more basil and freshly ground black pepper.