Preheat the oven to 400 degree Fahrenheit (200 degree Celsius).
Line a baking sheet with aluminum foil, and grease lightly.
Drain the tomatoes, reserving the juice for later.
Take each tomato and gently push about 2 or 3 slices of garlic in it, then arrange on the foil and drizzle with a little olive oil, generously salt and pepper and sprinkle with dried basil.
Roast in the top third of your oven for about 30 - 35 minutes, until beginning to char, but not drying out completely.
While the tomatoes are roasting, heat the oil and butter in a large soup pot, add the onion and carrot, a generous amount of salt and pepper, and saute over medium heat until the onion begins to lose it's shape, about 5 or 6 minutes.
Add the reserved tomato liquid, tomato paste, the balsamic vinegar, sugar, and stock, then increase the heat a bit and bring the soup to a low simmer.
When they are ready, add the tomatoes with their juices and continue to simmer the soup for about 30 minutes to an hour.
Remove the soup from heat, let cool for about 10 minutes, then whisk a few tablespoons of the hot soup into the sour cream to warm it, then add it and the basil to the soup and pulse-blending until smooth.
Serve with a crusty baguette and garnish with a little sour cream and a basil leaf or two.
Recipe Notes
The acidity in tomatoes can make the soup a bit bitter, but you counteract this with 2 spoons of sugar. Feel free to add another spoon if needed.
Make sure not to put the soup back on to the heat after adding the sour cream, or else it will curdle as it comes to a boil.
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