Heat the oven and line the pans. Set the oven to 350°F and line two baking sheets with parchment paper.
Whisk the dry ingredients. In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, pumpkin pie spice, and salt.
Beat the wet ingredients. In a large bowl, beat the brown sugar, vegetable oil, pumpkin puree, egg, and vanilla on low speed until smooth and combined, about 30 seconds.
Combine — and stop. Add the dry ingredients to the wet and mix on low just until no flour streaks remain. Overmixing makes tough cakes; a few lumps are fine.
Scoop and bake. Drop heaping tablespoons of batter onto the prepared sheets, about 2 inches apart — 12 mounds per sheet. Bake both sheets at once for 11 to 13 minutes, until the tops spring back when touched lightly.
Cool completely. Let the cakes cool on the sheets for 5 minutes, then move them to a rack and cool for 30 minutes more. Warm cakes melt the filling, so do not rush this.
Make the filling and sandwich. Beat the cream cheese and butter together until smooth and creamy, about 2 minutes. Beat in the powdered sugar, vanilla, and salt until light and fluffy. Spread about 2 tablespoons of filling on the flat side of half the cakes and top with the remaining cakes, flat side down. Chill for 30 minutes before serving so the filling sets.
