Adjust the oven rack to the center position and preheat the oven to 350 degrees F. Butter and flour a standard 12-cup Bundt pan or spray with Baking Spray with flour. Remove 3 tablespoons from the canned pumpkin puree and set aside.
In a large bowl, sift together the flour, baking soda, nutmeg, cinnamon, allspice, ginger, and clove. Set aside. In the stand mixer bowl, beat the oil and brown sugar on medium speed until light and fluffy.
Add the eggs, one at a time, beating well after each egg and scraping down the sides of the bowl. Add the pumpkin, apple cider, and vanilla and blend on low until thoroughly mixed. Gradually add the flour mixture and beat on low speed until just incorporated.
Pour the pumpkin batter into the prepared bundt and smooth the surface. Bake for 45 to 50 minutes, or until a cake tester or skewer inserted in the center comes out clean, and the edges start to pull away from the sides of the pan. Let cool in the pan for 10 minutes, then invert onto a rack and cool completely.
Meanwhile, make the Pumpkin Glaze: In a medium bowl, whisk together the powdered sugar, spices, pumpkin, 2 tablespoons half and half, and vanilla until smooth, adding more half and half as needed, 1 teaspoon at a time, until the glaze is pourable. Spoon the icing over the cake and allow to set for 20 minutes before slicing.