For the cake: Preheat oven to 350 degrees. Generously coat 3 (9-inch) round cake pans with shortening (tip: this works better than both cooking spray and butter) and lightly flour the pan, discarding the excess.
Sift together flour, baking powder, baking soda, cinnamon, and salt.
In the bowl of a stand mixer fitted with the beater blade, combine the sugar and canola oil on medium-low speed until well combined, 1 to 2 minutes. Add the pumpkin and vanilla until fully incorporated, about another minute. Beat in the eggs one at a time. Gradually add the flour mixture until just incorporated, being careful not to over mix.
Divide cake batter evenly into the 3 pans (I weigh the batter out for the best results). Rap each pan gently on the counter before putting in the oven so that the batter is evenly distributed.
Bake the cakes for 20 to 23 minutes, moving the pans around halfway through cooking time, until they are set and a toothpick comes out clean from the centers. Allow the cakes to cool in pans for 30 minutes. Run a knife along the sides of the pan and remove cake from pans. To do this, place a lightly-floured free hand or cardboard cake round directly onto the top of the cake in order to support the layer; then gently flip the cake pan over with your hand or cake round still supporting it, while gently tapping with the other hand until the cake slips out. Cool the cakes completely on wire racks.
For the frosting: With a hand mixer set at medium speed, beat the cream cheese, butter, and vanilla until smooth. Add the confectioners' sugar and continue to beat until light and fluffy, about 5 more minutes.
Assemble the cake: Place one cake layer on a cake plate or serving dish and spread with a layer of cream cheese frosting. Repeat with the second layer. Frost the top and sides of the cake with a crumb coat, then repeat with a thicker layer. The frosted cake will keep for up to three days in an airtight container.
Cook's Notes: The cake layers can be made up to three weeks in advance. If making three days in advance or less, just wrap each individual layer tightly in plastic wrap and tuck into a large sealed plastic sip-top bag. Store the cakes on the counter at room temperature. To store the layers longer than 3 three days (up to three weeks), wrap a layer of aluminum foil around the layer of