Preheat the oven to 350°F. Grease a 9×9 inch light metal baking pan and line it with parchment paper.

In a medium bowl, whisk together the all purpose flour, pumpkin pie spice, baking powder, baking soda, and salt; set aside.

In a large bowl, cream the softened butter and granulated sugar together for 2 minutes with an electric mixer on high speed.

Add in the eggs and vanilla; mix on medium speed until pale in color and smooth, about 1 minute.

Add the buttermilk and canned pumpkin puree; mix on medium-low speed until combined.

Add the dry ingredients to the wet ingredients and mix on low speed until combined, scraping the sides of the bowl as necessary.

Pour the batter into the prepared baking pan. Bake for 35-38 minutes, or until a toothpick inserted into the center comes out clean.

Let the cake cool in the pan on a wire rack for 30 minutes, then remove it from the pan and let it continue cooling on the rack.

While the cake cools, mix together the espresso, milk, sweetened condensed milk, pumpkin pie spice, and vanilla extract in a liquid measuring cup.

For the frosting, whip the butter on high speed until pale and fluffy, 5-10 minutes.

Add in the cold cream cheese and mix on medium speed until fluffy.

Sift in the powdered sugar a little at a time, mixing on low speed until combined, then on high speed for about 1 minute until fluffy.

Dissolve the instant espresso powder into the vanilla extract in a small ramekin, then add it to the frosting along with the pumpkin pie spice, mixing on medium-low speed until combined and then on high speed until light and fluffy.

When the cake is completely cooled, cut a very thin layer off the top. Poke holes on the top of the cake using a wooden stick.

Slowly pour the latte mixture on top of the cake, allowing it to absorb.

Frost the top of the cake with a thick layer of espresso cream cheese frosting. Sprinkle pumpkin pie spice on top and cut into 16 slices.