Position a rack in the center of the oven and heat the oven to 325 degrees F. Butter and flour a 10-cup Bundt pan; tap out excess flour.

In a large bowl with a hand mixer or in a stand mixer fitted with the paddle attachment, beat the butter and sugar on medium speed until well combined, about 1 minute. Add the oil and beat until combined, about 15 seconds. Add the eggs one at a time, mixing well on low speed. Add the vinegar and vanilla and mix again until just combined. Add half of the flour and the baking soda, salt, ginger, and nutmeg, mixing on low speed until just combined. Add half of the buttermilk and mix until just combined. Repeat with the remaining flour and buttermilk.

Stir the squash into the batter and transfer the batter to the prepared pan; smooth the top with a rubber spatula. Bake until a cake tester inserted in the center comes out clean, about 1 hour. Cool on a wire rack for 30 minutes; then carefully invert the cake onto the rack and remove the pan. When the cake is completely cool, transfer it to a serving plate.

In a medium bowl, whisk the sugar, buttermilk, vanilla, nutmeg, and salt until smooth. Add more buttermilk, a few drops at a time, as needed, until the icing is pourable but still quite thick. Pour the icing back and forth in thick ribbons over the cooled cake. Sprinkle the ginger on top. Let the icing set at room temperature, about 45 minutes, before serving.

Make Ahead: This cake can be made a day or two in advance. After the icing has set, wrap the cake in plastic and refrigerate. Let come to room temperature before serving.