Brown the butter. Add all 1 1/2 cups of butter to a saucepan over medium heat. Let it melt, then keep swirling as it foams and bubbles. In about 5 to 7 minutes you'll see golden brown bits on the bottom and it'll smell nutty and AMAZING. Pour it into a heatproof bowl right away, brown bits and all, so it doesn't burn.
Divide and cool. Measure 1 cup of brown butter into your mixing bowl for the cookies and put the rest in a separate bowl for the frosting. Let both cool. Pop the frosting butter in the fridge until it firms up to a soft, spreadable consistency like room-temperature butter.
Cook down the pumpkin. While the butter cools, add the pumpkin puree to a saucepan over low to medium heat. Stir constantly for about 10 minutes to cook out the extra moisture. It will shrink a little bit, and that's totally okay because it's accounted for! Take it off the heat and let it cool.
Mix the wet ingredients. Beat the 1 cup of brown butter with the brown sugar and granulated sugar for about 2 minutes, until fully combined. Add the eggs, vanilla, and cooked-down pumpkin and mix again until smooth.
Add the dry ingredients. In a separate bowl, whisk together the flour, oats, cinnamon, pumpkin spice, baking powder, and salt. Fold them into the wet ingredients just until no dry streaks remain.
Chill the dough. Cover and refrigerate for 30 minutes. While it chills, preheat your oven to 375°F and line your baking sheets with parchment paper.
Scoop. Scoop the dough into 24 equal balls, a little over 2 tablespoons each, and space them out on your baking sheets.
Bake. Bake at 375°F for 11 to 13 minutes, until the edges are set and the centers still look a little underbaked. They'll keep baking on the pan for about 10 minutes after they come out.
Shape them up. Right out of the oven, place a biscuit cutter over each cookie and wiggle it around to make them perfectly round and the same size. This makes your cream pies look SO pretty when you sandwich them.
Cool. Let the cookies cool on the pan for 30 minutes, then transfer them to a wire rack to cool completely.
Make the frosting. Beat the cream cheese, the remaining brown butter, vanilla, and brown sugar for 2 minutes, until smooth and creamy. Add the powdered sugar and heavy cream and beat until light and fluffy. Transfer to a piping bag.
Assemble. Flip half the cookies upside down. Pipe a generous layer of frosting on each one, add a drizzle of salted caramel, then top with the remaining cookies and gently press to sandwich. And ENJOY!