Preheat the oven to 400 degrees. In a large cast-iron skillet, heat 2 tbsp. oil over medium-high. Add half the sprouts and cook, cut side down, until browned, about 3 minutes. Transfer to a bowl; season with salt and pepper. Repeat with the remaining oil and sprouts.
In a large nonstick skillet, melt 3 tbsp. butter over medium-high. Add the shallots, garlic and thyme; season. Cook, stirring often, until golden, about 5 minutes. Stir in 1/2 cup cider; cook over medium heat, stirring often, until the shallots are deep brown and most of the liquid has evaporated, about 10 minutes. Transfer to a medium bowl.
Wipe out the skillet; add the remaining 1 cup cider, 3 tbsp. butter and the vinegar. Cook over medium until the sauce resembles caramel sauce, about 10 minutes. Pour into a 9-inch pie dish, swirling to coat the bottom. Arrange half the sprouts in the dish, cut side down, in a single layer. Fill the crevices with about two-thirds of the shallot mixture. Repeat with the remaining sprouts and shallots. Top with the pie dough, tucking the edges into the dish. Pierce the dough all over with a fork. Bake until golden, 20 to 25 minutes. Let cool 5 minutes. Run a knife around the edge of the dish. Invert the tart onto a platter. Cut into wedges and serve with the creme fraiche.