Make the cakes
Set the oven rack in the lower third of the oven. Preheat the oven to 400°F. Line two baking sheets with parchment paper or nonstick silicone baking mats.
In a medium bowl, sift together the flour, baking soda, baking powder, salt, cinnamon, nutmeg, and cloves; set aside.
In the bowl of a standing mixer fitted with the paddle attachment, combine the butter, brown sugar, and granulated sugar and beat on medium-high speed until light and fluffy, about 2 minutes, scraping down the sides of the bowl halfway through. Beat in the oil, egg, and pumpkin puree, beating until blended after each addition. With the mixer on low speed, beat in the flour mixture in three batches, alternating with the milk.
Scoop 8 scoops of batter with a #40 ice cream scoop or 2 tablespoons’ worth, about 2 inches apart, onto each of the prepared baking sheets. Baking one pan at a time, bake the cakes, rotating the sheet about two-thirds of the way through the baking time, until a cake tester inserted into the center of the cakes comes out clean, 8 to 10 minutes. Remove the baking sheets to cooling racks and let the cakes cool for 10 minutes. Then remove to the rack with a metal spatula and allow to cool completely.
To make the cream
In the bowl of a standing mixer fitted with the paddle attachment, combine the butter, superfine sugar, cinnamon, and salt and beat until light and fluffy, and until the sugar has dissolved and is no longer gritty, 5 to 10 minutes. Scrape down the sides of the bowl about halfway through. Beat in the vanilla. Transfer to a small bowl; set aside.
In a small saucepan, whisk together the flour and milk until smooth. Cook over medium heat until the mixture forms a very thick paste and the flour taste has cooked out, 6 to 10 minutes.
Transfer to the mixer bowl and beat until completely cool. Beat in the butter-sugar mixture on medium-high speed until light and fluffy, 4 to 5 minutes.
Assemble the pies
With a small spatula or butter knife, spread the flat sides of 8 of the cooled cakes with about 2 Tbs. of the cinnamon cream. Top with the remaining 8 cakes.
Photo: Ben Fink