Preheat the oven to 325. In a small saucepan, combine the water with the sugar, vinegar and cinnamon and simmer for 5 minutes, stirring until the sugar dissolves. Add the currants and simmer for 3 minutes, then remove from the heat.
Meanwhile, line a baking sheet with aluminum foil. Rub the squash with the olive oil and season with salt and pepper. Using a fine grater, grate the ginger over the cut sides of the squash and rub it into the flesh.
Transfer the squash to the prepared baking sheet, cut side down, and roast for about 15 minutes, until the squash starts to soften. Turn the squash cut side up and roast for about 17 minutes longer, until tender; transfer to a work surface and let cool slightly.
Increase the oven temperature to 425. Cut the squash into 1 1/2-inch-thick wedges. Lightly rub the foil with oil and arrange the squash on the baking sheet; roast for about 25 minutes, turning once halfway through, until golden and crisp along the edges.
Arrange the squash on a platter. Discard the cinnamon stick from the currants and add the currants to the squash. Drizzle with the pickling liquid and crème fraîche and serve.