How to Make It
Preheat the oven to 250°. Drizzle the pure olive oil on a rimmed baking sheet and sprinkle with the thyme and a pinch of salt. Arrange the tomatoes, cut side down, on the sheet and roast for 1 hour. Pull off the skins. Turn the tomatoes and roast for 30 minutes longer, until slightly dried. Let cool, then coarsely chop.
Meanwhile, in a large pot of cold water, bring the potatoes to a boil. Cook over moderately high heat until tender, about 20 minutes. Drain and let cool. Peel the potatoes and quarter them. Transfer the potatoes to a large bowl and toss with 1 tablespoon of vinegar.
Bring a medium saucepan of salted water to a boil. Add the haricots verts and boil until tender, about 5 minutes. Drain and rinse under cold water; pat dry.
On a work surface, mash the garlic to a paste with a pinch of salt. Transfer the garlic paste to a mini food processor, along with the mayonnaise and the remaining 1/2 tablespoon of vinegar, and puree until smooth. Add the extra-virgin olive oil in a thin stream and process until blended. Add the tomatoes and pulse several times, stopping while the dressing is still slightly chunky. Season with salt and pepper.
Set a medium skillet over high heat. Add the sausage and cook, turning once, until browned and crisp, about 2 minutes. Using a slotted spoon, transfer the sausage to the potatoes. Add the beans, basil and tomato dressing, season with salt and pepper and toss well. Transfer to a bowl and serve.