To a 10-inch cast-iron skillet set over medium-high heat, add the butternut squash and milk. When the liquid begins to boil, turn the heat to medium-low and cook until soft, about 10 minutes. Drain the squash (discard the milk), then whisk it into a purée. Scrape the purée into a small bowl and set aside. Clean the skillet and return it to the stove.
Wash the delicata squash, cut it in half lengthwise, and discard the seeds, then slice it into ½-inch half-moons. To the empty skillet, add the sugar and ½ cup of water. Turn the heat to medium-high and cook, stirring occasionally, until the mixture begins to boil and is starting to brown, about 6 minutes. Stir in ¼ cup of water (the mixture will bubble vigorously), then arrange the delicata squash atop the caramel in one tight layer (reserve any pieces that don't fit for another use). Cook the squash in the caramel, turning once and adding water by the ¼ cup if the pan looks dry, until tender and browned on both sides, 16–20 minutes. Lightly press the squash into a single layer, grease the sides of the pan with butter, and set aside.
Position a rack in the center of the oven and preheat to 350°F. Into a medium bowl, sift the flour, baking powder, salt, ginger, and nutmeg, and whisk to combine. In the bowl of a mixer fitted with the paddle attachment, beat the butter and turbinado sugar on medium-high until combined, about 1 minute. Beat in the maple syrup, followed by the eggs one by one, scraping down the bowl after each addition. Beat in the cider and squash purée, then add the flour mixture and beat on low speed until just combined.
Pour the batter over the squash, spreading it to the edge of the skillet. Bake until deep golden and a toothpick inserted in the center comes out clean, 40–45 minutes. Run the tip of a knife around the rim of the skillet, then carefully invert the pudding onto a large plate. Serve warm or at room temperature.