Preheat the oven to 400°.
Peel each shallot and cut in half the long way through the root. With double shallots (ones with two bulbs), peel and slice both.
Set the skillet over medium high heat and sprinkle the sugar so that it coats the bottom of the pan in an even layer. Let the sugar cook without stirring until the edges begin to caramelize. Wearing a hot pad, swirl the pan gently to help the sugar caramelize evenly.
While the sugar caramelizes, gently roll the puff pastry until an even 1/8-inch thick.
Once the caramel reaches a golden brown, turn the heat to low. Add in the butter, swirling until melted. Stand away from the pan to avoid any caramel that spits and swirl in the balsamic vinegar. Lay the thyme sprig in the center of the pan.
Working in a circular pattern from the outside in, lay the shallots cut side down in the caramel. Wedge the shallots in as tightly as possible while keeping them slightly submerged. Finish with one larger shallot right in the center. Lay your puff pastry over the shallots, gently tucking the edges down into the skillet. Continue cooking over low heat for about 5 more minutes, then place in the hot oven. Bake until the pastry is a dark golden brown, about 35–40 minutes.
Remove the skillet from the oven and let sit for one minute only to settle the caramel. If the tart stays in the pan any longer, the caramel will harden and the shallots will stick to the pan. Place a large plate over the skillet and invert plate and skillet both to remove the tart from the pan. Sprinkle with sea salt to taste and serve warm or at room temperature.
Yields 6-8 servings