How to Make It
Preheat the oven to 225°. In a medium bowl, toss the plum tomatoes with 1/2 cup of the olive oil, the thyme, garlic, sugar and 2 teaspoons of kosher salt; season with white pepper. Spread the tomatoes out on a rimmed baking sheet and bake for 1 hour and 20 minutes, turning them over halfway through cooking. Let cool. Transfer to a bowl and refrigerate until chilled.
Meanwhile, snap off the tough outer leaves of the artichoke until you reach the pale green leaves. Using a sharp knife, cut off the top two-thirds of the leaves, then peel the base and stem. Using a spoon or melon baller, scoop out the hairy choke. Cut the artichoke into 1/2-inch pieces.
Squeeze the tomato halves to soften them, then scoop the seeds from the tomatoes and discard. Cut each half into 4 quarters. In a large saucepan, heat the remaining 1/4 cup of olive oil. Add the onion, shallot, green pepper and artichoke, season with salt and cook over moderate heat until the vegetables soften, about 10 minutes. Add the green tomatoes, cover and cook over moderately low heat, stirring occasionally, until softened and most of their liquid has evaporated, about 1 hour.
Transfer the vegetables to a blender and puree until smooth. Strain the puree through a fine sieve set over a medium bowl and refrigerate until chilled. Stir in the cream and water, season with salt and white pepper and refrigerate.
Drain the plum tomatoes and pat dry with paper towels. Finely chop the tomatoes and transfer to a bowl. Add 2 tablespoons of the basil and season lightly with salt and white pepper. In another bowl, season the pear tomatoes with salt and white pepper.
Place a 3-inch round ring mold or biscuit cutter in the bottom of each of 4 shallow bowls. Press the plum tomato mixture into the molds, then carefully remove the molds. Mound the pear tomatoes on top. Pour the soup all around, garnish with the remaining 2 tablespoons of basil and serve.