Preheat the oven to 350 F (177 C). Grease and line the bottom of a 9- inch (23 cm) springform or cake pan with parchment paper.
In a large bowl, whisk together the flour, baking powder, baking soda, ground cinnamon, ginger, nutmeg, cloves and salt together until well combined.
2 cups (240 g) all purpose flour, 2 teaspoons baking powder, ½ teaspoon baking soda, 2 teaspoons ground cinnamon , 1 teaspoon ground ginger, ½ teaspoon ground nutmeg, ½ teaspoon ground cloves or all spice, ½ teaspoon salt
In a large mixing bowl fitted with whisk attachment whisk the eggs, sugar and light brown sugar together on low speed until combined. Increase the speed to high and whip for 4 - 5 minutes. The egg and sugar mixture will be thick and lighter in color.
3 large eggs, room temperature, 1 cup (200 g) granulated sugar, ½ cup (100 g) light brown sugar
With the mixture on medium speed, stream in the olive oil. Once all the olive oil is streamed in, increase the speed to high for 1 minute. Scrape the sides and bottom of the mixing bowl as needed. On low, whisk in the pumpkin puree and vanilla, followed by the dry ingredients. Mix until just incorporated.
¾ cup (180 ml) olive oil, 1 cup (260 g) pumpkin puree, 2 teaspoons (10 ml) vanilla extract
Evenly pour the pumpkin spiced olive oil cake batter into the prepared pan. Bake the olive oil pumpkin cake for 45 - 50 minutes or until a toothpick inserted into the center of the cake comes out clean or with a few moist crumbs. The crust will be golden brown and the top should bounce back when gently pressed.
Cool the cake in the pan for 10 minutes. Gently release the side of the springform pan and allow the cake to cool for 45 - 60 more minutes or until room temperature before topping with frosting or glaze.
While the cake cools, make the cinnamon cream cheese frosting. Add the room temperature cream cheese and butter to the bowl of a stand mixer fitted with paddle attachment. Cream together the butter with the room temperature cream cheese until just combined, about 2 - 3 minutes. Add the cinnamon and vanilla extract. Mix until creamy and well combined, about 2 - 3 more minutes on low speed. Stop and scrape down sides of bowl as needed.
4 oz. (113 g) cream cheese, room temperature, 4 tablespoons (57 g) unsalted butter, room temperature, 1 - 2 teaspoons ground cinnamon , ½ teaspoon vanilla extract
Mix in the sifted powdered sugar ½ cup (60 g) at a time, mixing in each addition of powdered sugar until incorporated. Scrape down sides of mixing bowl as needed. Once all the sugar has been added, mix on high until the cream cheese cinnamon buttercream is fully incorporated and your desired consistency.
2 cups (240 g) powdered sugar
Transfer the pumpkin cake to a serving plate or serve right from the pan. Top with the frosting, glaze or a dusting of powdered sugar and enjoy.