Preheat the oven to 350°F (175°C). Grease a 9×9-inch square baking pan and line it with parchment paper, leaving some overhang for easy removal.
Whisk together the flour, sugar, pumpkin pie spice, baking powder, baking soda, and salt in a large bowl. In a separate bowl, whisk the pumpkin puree, oil, egg, yogurt, and vanilla until smooth.
Add the wet ingredients to the dry ingredients and gently fold until just combined. Avoid overmixing to keep the cake soft and tender.
Transfer about ½ cup of the batter to a small bowl and stir in the espresso powder. Spread the remaining pumpkin batter into the prepared pan, dollop the espresso batter over the top, and gently swirl it through with a knife.
Bake for 20 to 25 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan for about 20 minutes, then transfer it to a wire rack to cool completely.
While the cake cools, combine the espresso, cream, and brown sugar for the latte soak. Stir until the sugar dissolves. Once the cake is cool, poke small holes across the surface and brush the coffee soak evenly over the cake.
Beat the softened butter and powdered sugar until fluffy. Add the cream, espresso powder, and vanilla and beat until smooth and creamy. Spread the frosting over the cooled cake, drizzle with a little reserved coffee soak, and finish with a light sprinkle of cinnamon.