Preheat the oven to 350F. Grease and line an 8x8" baking dish with parchment paper.
Place the brown sugar, pumpkin puree, butter, oil, greek yogurt, eggs, and vanilla extract in a large bowl. Whisk until combined and smooth.
Add the flour, pumpkin pie spice, baking soda, baking powder, and salt. Gently stir with a rubber spatula until just combined, until no streaks of flour remain.
Pour the batter into the prepared baking dish and spread it out so it fills the pan evenly. Transfer to the oven to bake for 20-30 minutes, until an inserted toothpick comes out clean. Let it cool in the pan for 5 minutes, then remove and place on a cooling rack to cool completely.
When the cake is almost cooled, make the frosting. Add the cream cheese and butter to a medium bowl. Use a hand mixer or stand mixer to mix until well combined, about 2 minutes.
Add the powdered sugar, vanilla, and salt and mix until smooth, about 2 minutes.
When the cake is completely cooled, spread the frosting on top of the cake, or pipe it on in whatever pattern you'd like. Enjoy!