Preheat the oven to 350 F. Line and grease a square metal 8 or 9” pan with parchment paper.
Brown the butter: In a saucepan, melt the butter then turn the heat to medium high and cook the butter as it spits and sputters, foams and then ‘browns’: you’ll see brown bits at the bottom of the pan, the butter will go mostly quiet and you’ll smell a nuttiness.
Immediately pour the butter into a heatproof bowl, making sure to scrape the little brown bits into the bowl (that’s where the flavor is!!). Add the sugar, salt, vanilla and pumpkin spice and whisk for a few minutes, until it’s glossy.
Add the egg and yolk and whisk aggressively for a few minutes until smooth.
Whisk in the pumpkin puree, sour cream and flour. Pour the batter into the prepared pan and bake for 25-30 minutes until a cake tester comes out mostly clean. Let cool completely.
For the buttercream: the cream cheese should be very soft. Use the back of a rubber spatula to press it into a bowl and smooth out any lumps. Add the remaining ingredients and beat until smooth. Spread over cooled bars. Add a couple of dollops of caramel and swirl it into the buttercream.