To roast tomatoes: Line a large rimmed baking sheet with parchment paper. Preheat oven to 300°F. Toss tomatoes in a bowl with thyme, oil, salt and pepper. Arrange on the prepared baking sheet, cut-side down. Roast for 11⁄2 hours. Turn each tomato cut-side up, top with garlic and roast until the tomatoes are slightly shriveled and browned at the edges, 30 to 45 minutes more. When cool enough to handle, remove and discard the tomato skins.
To prepare tomato butter: Combine 1⁄4 cup of the roasted tomatoes, butter, basil and vinegar in a food processor; pulse until pureed, scraping down the sides as needed. Leave at room temperature if using right away. (Otherwise,
refrigerate for up to 1 week or freeze for up to 6 months. Refrigerate the remaining roasted tomatoes, covered with a layer of olive oil, for up to 1 week or freeze for up to 6 months.)
To prepare oysters: Position an oven rack about 4 inches from the broiler; preheat broiler to high. Spread a layer of rock salt (or a layer of crumpled foil) on a large rimmed baking sheet to keep the oysters level. Shuck oysters, leaving them on the half shell. Nestle the oysters, in their shells, in the salt (or foil). Top each oyster with desired amount of the tomato butter. Broil, watching closely, until the butter is bubbling and the edges of the oysters begin to curl, 5 to 10 minutes, depending on the size of the oysters.