How to Make It
Preheat oven to 425°F. Toss together tomatoes, onion, garlic, thyme, oil, pepper, and 1⁄2 Tbsp. of the salt on a large rimmed baking sheet. Arrange tomatoes evenly, cut side down, on baking sheet. Bake in preheated oven until tomato skins begin to blister, 25 to 30 minutes.
Peel and discard skins from cooked tomatoes; discard thyme from baking sheet. Transfer peeled tomatoes, onion, garlic, and any accumulated juices from baking sheet to a Dutch oven. Add broth, vinegar, and remaining 1⁄2 Tbsp. salt. Process mixture with an immersion blender until smooth, about 2 minutes. Bring to a simmer over medium; simmer 10 minutes.
Remove from heat; stir in cheese until melted and smooth. (If soup is not completely smooth, process once more with immersion blender.) Spoon soup evenly into 6 bowls; sprinkle with additional cheese, pepper, and chives.