Spray a 9x13-inch sheet pan with nonstick spray and line with parchment. Preheat the oven to 350°F.

Whisk together flour, baking powder, baking soda, pumpkin pie spice, cinnamon, and salt in a large bowl.

In another bowl, whisk olive oil, granulated sugar, brown sugar, eggs, pumpkin puree, and vanilla extract.

Add the wet mixture to the dry and stir until just combined. Pour into the prepared pan and bake for 35–40 minutes, or until a toothpick inserted in the center comes out clean.

Remove from the oven and invert onto a parchment-lined cooling rack. Let cool completely.

Whisk together eggs, milk, sugar, cornstarch, and salt in a saucepan until smooth. Cook over medium-high heat, stirring constantly, until thickened.

Pour into a bowl, press plastic wrap directly on the surface, and let cool to room temperature.

In a stand mixer, whip room-temperature butter until light and fluffy. Scrape in the vanilla bean (or add vanilla bean paste), then beat in the cooled pastry cream until smooth.

Slice the cooled cake into three equal rectangles. Spread or pipe ⅓ of the frosting onto the first layer, top with the second piece of cake and another ⅓ of the frosting, then finish with the last layer and the remaining buttercream. Leave the sides unfrosted for a rustic look. Chill for at least 1 hour before serving.