Preheat the oven to 325 degrees. Rub the cut sides of the squash with 1 tablespoon of olive oil and season with sea salt and pepper. Rub the ginger over the cut sides of the squash and into the flesh. Lay squash on a baking sheet, cut side down, and roast for 15 minutes, until it starts to soften. Turn squash, cut side up, and roast for 17 minutes longer, until tender; then transfer to a work surface and let cool slightly. Increase the oven temperature to 425 degrees. Cut the squash into 1 1/2-inch-thick wedges. Oil the baking sheet with 1 tablespoon of olive oil and arrange squash and apples on it; roast for 25 minutes, turning halfway through, until golden. Line a platter with arugula and arrange squash and apples on top.