Preheat the oven to 350ºF (180ºC). Line a 6-serving shallow jumbo muffin pan with jumbo muffin liners.* (The jumbo muffin 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 will also work.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger and nutmeg.
In a large bowl, whisk together the sugar, oil, eggs and vanilla until smooth and well combined. Mix in the pumpkin purée. Stir in the flour 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 23-27 minutes. Let the cakes cool in the pan for 5 minutes, then transfer to a cooling rack.
In a small bowl, whisk together the brown sugar, flour, cinnamon and salt. Add the melted butter and whisk until well combined.
Transfer the mixture to a piping bag fitted with a 1/4-inch wide round tip.
Vanilla Bean Icing
In a small bowl, whisk together the butter, vanilla bean paste, powdered sugar, meringue powder, salt and 1 tablespoon of milk until well combined. Whisk in more milk as needed to reach your desired consistency.
Brush the icing on the cooled cakes with a small pastry brush or make the icing a little thinner and drizzle the icing on if you prefer.