Core and roughly chop the heirloom tomatoes.
In a large glass bowl* combine the tomatoes, garlic and salt. Mix together until well combined.
Let this mixture marinate for 30-60 minutes, chilled or covered on the counter.
Pass the tomato mixture through a food mill with a fine mesh plate, or push them through a fine mesh strainer. This will remove most of the skins and seeds.
In a small bowl, combine the champagne vinegar with the extra virgin olive oil. Add the dressing mixture to the tomatoes and season to taste with salt and freshly ground black pepper.
To adjust seasonings, if necessary, add more salt, freshly ground black pepper, extra virgin olive oil or champagne vinegar.
Chill thoroughly, in the freezer if possible, for 1-2 hours.
In a small bowl, whisk the Greek yogurt to a smooth consistency. Adjust seasoning to taste.
Finely chop the Thai basil and parsley leaves.
Evenly divide the Greek yogurt into chilled serving bowls and sprinkle the Thai basil leaf slivers around the yogurt.
Spoon the chilled tomato soup around the yogurt.
Garnish with the parsley leaves.