In a small saucepan, bring 1 1/2 cups stock and the sundried tomatoes to a low boil. Lower the heat and simmer until softened, 6 to 7 minutes.
Meanwhile, in a large saucepan, heat the EVOO, 2 turns of the pan, over medium-high heat. Add the onion, carrot, garlic, bay leaf and thyme; season with salt and pepper. Cook until the veggies are softened, about 5 minutes. Stir in the vodka; deglaze and reduce for a minute. Add the crushed tomatoes and reduce to thicken a bit, 15 to 18 minutes. Add the remaining 2 1/2 cups stock, the reconstituted sun-dried tomatoes and their stock, and the basil. Puree with an immersion blender, return to a bubble and stir in the cream.
Dip the rims of mugs, bowls or small sipping cups in warm water, then dip in the rim rub. Fill the vessels with the soup. (For parties, serve in shot glasses.)