Preheat oven to 400°F. Place carrots on a baking sheet. Toss with olive oil and season with sea-salt. Cook in the oven for 30 minutes or until they begin to caramelize.
While the carrots are roasting, add the sugar to the water in a small saucepan. Cook until a dark caramel color, about 4-5 minutes. Remove from the heat and add the vinegar and a pinch of salt. Swirl to combine. Immediately pour the caramel into the bottom of a 9-inch pie pan and swirl to evenly coat the bottom. Arrange the carrots, cut side down, on top of the caramel. Some of the carrots may need to be trimmed in order to fit
Reduce the oven temperature to 375°F. Sprinkle the thyme and goat cheese evenly over the carrots. Flour a work surface and roll out the pie dough into a 12-inch circle. Place the pie crust on top of the carrots and fold under the edges so that it fits snuggly into the pie tin. Using a skewer or fork, poke holes all over the crust. Bake in the oven for 55-65 minutes or until the crust is cooked through.
Immediately invert onto a platter or cutting board. If you let it sit for too long, the caramel will stick to the pan and you may not be able to get the tarte tatin out in one piece. Let cool for 20 minutes. Slice into 8 pieces and serve with a bitter greens salad.