In a large saucepan over medium heat, warm a tablespoon of the olive oil. Add the garlic and sauté, stirring constantly for 1 minute. Add 2 tablespoons of the butter, the red onion, and thyme. Increase heat to medium high and cook for 7-10 minutes, stirring frequently. Add 1 tablespoon sugar and stir frequently as onion caramelizes, 2-3 minutes. Stir in the balsamic vinegar, reduce heat to low, and stirring occasionally, simmer for 35-40 minutes, until thick and jam-like. Set this onion jam aside to cool.

Preheat the oven to 375˚F. Butter a 8" or 9" round baking dish or casserole.

Cut large shallots in half lengthwise, and leave the smaller ones whole. Heat remaining 3 tablespoons of olive oil in a large skillet over medium-high heat. Gently add the shallots to the skillet. Sauté until golden, 6-8 minutes, occasionally gently turning them. Try to keep the shallots as intact as possible. Sprinkle the last tablespoon of sugar over the shallots and cook 3-5 minutes. Add the remaining butter. Continue to cook, stirring occasionally, until shallots are a deep caramel color, about 5 minutes. Drizzle in the red wine vinegar, reduce heat to low, sprinkle with salt to taste, and cook until shallots are soft, about 20 minutes.

Transfer the shallots to the baking dish, flat sides up. Spoon the caramelized juices over them, then spread the onion jam over the shallots. Place the sheet of puff pastry on top, tucking the edges in around the shallots. Bake until golden and crisp, 12-15 minutes. Allow to cool 5 minutes, then run a sharp knife around the perimeter of the baking dish. Place an inverted serving platter on the top of the baking dish, and quickly and carefully flip the tart over onto the platter. Serve warm.