Put 1/2 teaspoon herb vinegar in a tiny bowl or ramekin. Add the cream and stir; set aside to thicken.
Light your bbq grill — charcoal or gas. Roast the eggplant for 10-12 minutes, turning every few minutes until the skin is charred and the flesh is soft.
Place roasted eggplant in a bowl and cover tightly; let them steam for 5 minutes. Cut the eggplant in half lengthwise and scoop out the flesh, placing it in a food processor. (I prefer using a tall, narrow bowl and an immersion blender.)
Add 1/2 teaspoon of herb vinegar, the cream/vinegar mixture, thyme, cinnamon, and salt. Blend until smooth. With the blender motor running, add olive oil a little at a time until the eggplant mixture is creamy and smooth.
Whisk the remaining vinegar, sugar, and a pinch of salt in a large mixing bowl. Toss in both kinds of tomatoes and tumble to coat with the vinegar.
To serve, divide the purée among four (4) dinner plates and top with tomatoes (use a slotted spoon so they drain well). Crumble feta over the top and sprinkle with black sesame seeds and herb leaves. Add an extra drizzle of olive oil, if desired.