Brown the butter. Add the butter to a saucepan over medium heat. Cook, stirring frequently, as it melts, foams, and begins to turn golden. Continue cooking until you see deep golden-brown specks on the bottom and the butter smells nutty.

Immediately pour the brown butter, including all those toasted milk solids, into a large mixing bowl. Let cool for 10–15 minutes.

Add the brown sugar and granulated sugar to the slightly cooled brown butter and whisk until thoroughly combined.

Add the egg, egg yolk, and vanilla. Whisk until the mixture looks smooth and glossy.

Add the pumpkin puree and whisk until completely incorporated.

In a separate bowl, whisk together the flour, cinnamon, pumpkin pie spice, baking powder, baking soda, and salt.

Add the dry ingredients to the pumpkin mixture and gently fold until just combined. Do not overmix.

Spread the batter evenly into the prepared pan. The batter should be fairly thick.

Bake for 25–32 minutes, or until the edges are set and a toothpick inserted into the center comes out with a few moist crumbs. Don't wait for a completely dry toothpick.

Let the bars cool completely in the pan before frosting.

Make the cream cheese frosting. Beat the softened cream cheese and butter together for 2–3 minutes, until completely smooth and creamy.

Add the powdered sugar, vanilla, and salt. Beat until fluffy and smooth. If needed, add 1–2 teaspoons of heavy cream or milk to reach your desired consistency.

Spread the cream cheese frosting generously over the completely cooled pumpkin bars. Dust lightly with cinnamon or pumpkin pie spice, if desired.

For the cleanest slices, refrigerate the frosted bars for about 20–30 minutes before lifting them from the pan and cutting into squares.