Preheat oven to 400°F. Peel and cut shallots in half lengthwise, keeping a bit of root intact to hold together; set aside.

Melt 2 tablespoons butter in a 10-inch ovenproof stainless steel skillet; remove skillet from heat. Tightly arrange shallots, cut sides down, in skillet. (Skillet should be fully covered; better to be overcrowded than under.) Return skillet to heat over medium; sprinkle shallots with salt. Cover skillet using a tight-fitting lid; cook, undisturbed, until shallots begin to soften and become translucent, about 5 minutes. Uncover and cook, undisturbed, until cut sides are a rich golden brown, 2 to 3 minutes, rotating skillet as needed to encourage even browning.

Add vinegar and 4 teaspoons water to skillet; cook over medium, undisturbed, until liquid is almost fully absorbed, about 3 minutes. Carefully transfer shallots to a plate using a spatula to keep shallots tightly together, and set aside. Clean and dry skillet.

Return skillet to heat over medium. Add sugar in an even layer; cook, swirling skillet occasionally, until sugar is melted and amber-colored, about 8 minutes. Remove skillet from heat; add remaining 1 tablespoon butter, stirring constantly, until well combined. Quickly and carefully, tightly arrange shallots, cut sides down, over caramel in skillet. Cover with puff pastry sheet, tucking excess corners under pastry; bake until puffed and golden, about 20 minutes.

Whisk together egg and remaining 1 tablespoon water in a small bowl; brush pastry with egg wash. Continue baking until deeply golden, about 5 minutes.

Immediately invert Tatin onto a large plate while hot. Let cool for 15 minutes before serving. Garnish with flaky sea salt and herbs; serve with crème fraîche, if using.