Preheat oven to 400°F (200°C). Line a rimmed baking sheet with foil.

On prepared pan, toss together carrot, sweet potato, parsnip, garlic, oil, 1 teaspoon (3 grams) salt, and pepper. Spread into an even layer.

Bake until vegetables are fork-tender, 8 to 10 minutes. Leave oven on.

Meanwhile, in a small saucepan, stir together sugar and 3. tablespoons (45 grams) water. Cook, without stirring, over medium-high heat until amber-colored, 10 to 14 minutes. Remove from heat, and carefully add vinegar and remaining ¼ teaspoon salt, stirring until combined. Quickly pour caramel into a 9-inch glass pie plate; rotate plate to evenly coat bottom. Arrange carrot, sweet potato, and parsnip in concentric circles, overlapping, on top of caramel. (Use smaller carrot and parsnip pieces to cover any gaps. It’s OK if you have a few vegetable pieces left over.) Sprinkle with chopped sage.

On a lightly floured surface, roll Puff Pastry Crust into a 10-inch circle (about ¼ inch thick). Drape dough over pie plate. Trim about ¼ inch off edges, and tuck edges into plate and around vegetables. Prick dough all over with a fork in ½-inch intervals.

Bake until crust looks dry and has a slightly golden color, 16 to 18 minutes. Reduce oven temperature to 375°F(190°C), and bake until crust is golden brown, 10 to 12 minutes more. Let cool in plate for about 5 minutes. Invert tart onto a large serving plate. Garnish with goat cheese and sage leaves.