Preheat the oven to 400°F. Line a rimmed baking sheet with aluminum foil.
Stem the tomatoes and cut each in half lengthwise. Remove seeds if you prefer (gently squeeze or use a spoon). Arrange the tomatoes cut side up in a single layer on the prepared baking sheet. Add the 8 smashed garlic cloves and the 8 thyme sprigs to the sheet. Drizzle with the 1 tablespoon extra virgin olive oil and toss/lightly coat the tomatoes and garlic.
Roast the tomatoes, garlic, and thyme for 40–50 minutes, until the tomato skins soften and begin to burst. Remove from the oven and discard the thyme sprigs. Let the roasted tomatoes and garlic cool slightly.
In a large stockpot over medium heat, melt the 2 tablespoons butter. Add the shredded carrot (about 1 cup) and the shredded ½ medium yellow onion (about ¾ cup). Sprinkle in ¼ teaspoon of the kosher salt and sauté, stirring often, for about 5 minutes or until the vegetables are softened.
Add the 1 tablespoon tomato paste and the ½ teaspoon cayenne pepper to the pot. Cook, stirring, for 1 minute.
Add the roasted tomatoes and all of the roasted garlic to the pot, then pour in the 1½ cups vegetable broth. Cook for about 5 minutes to combine flavors.
Blend the soup until smooth: either use an immersion blender directly in the pot, or carefully transfer the hot mixture in batches to a blender (vent the lid and hold a towel over it) and blend until smooth. Return the blended soup to the pot if needed.
Stir in the 1 cup half-and-half. Reserve 2 tablespoons of the ½ cup slivered fresh basil for garnish, and stir the remaining basil into the soup. Warm the soup over medium heat for about 5 minutes more; do not boil.
Taste and add the remaining ¾ teaspoon kosher salt (you used ¼ teaspoon earlier), adjusting as needed but not exceeding the total 1 teaspoon called for.
Divide the soup among serving bowls. Evenly distribute the 1 cup cooked orzo pasta among the bowls (about ¼ cup per bowl if serving four). Sprinkle the reserved basil and the 4 tablespoons grated Parmesan cheese evenly over the bowls and serve hot.