Preheat the oven to 350ºF (180ºC). Line a 6-serving shallow jumbo muffin pan with jumbo muffin liners.* (The liners will fit by pressing them in around the inside bottom edges.) The muffin pan I used has cups that are 4-inches across by 1-inch deep. A hamburger bun pan for 4-inch buns or six 4-inch cake pans also work.
In a medium-sized bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger and nutmeg until well blended.
In a large bowl, whisk together the sugar, oil, eggs and vanilla until smooth and creamy. Mix in the pumpkin purée. Stir in the flour mixture in two additions, just until combined.
Transfer the batter to the muffin cups, evenly dividing the batter among the cups. Pipe the cinnamon swirl mixture in a spiral on the top of the batter of each one.
Bake until a toothpick inserted into the center of the cake part comes out clean, about 22-27 minutes. Let the cakes cool in the pan for 5 minutes, then transfer them to a cooling rack.