1. Place butter in a large, light-colored pan over medium heat. Stir constantly until melted, foamy, and golden brown with a nutty aroma, about 5-6 minutes. Immediately pour into a heatproof bowl and cool in refrigerator for 30 minutes.
2. Preheat oven to 350°F. Line 2 large baking sheets with parchment paper or silicone mats.
3. In a large bowl, beat 1/2 cup room-temperature brown butter with brown sugar on medium-high speed until light and fluffy, about 2 minutes.
4. If using pumpkin, blot with paper towel to remove excess moisture. Add pumpkin puree (or egg) and 1 tsp vanilla to the butter mixture. Mix until combined.
5. Add flour, oats, baking soda, cinnamon, and 1/2 tsp salt. Fold together with a spatula until just combined. Dough will be thick.
6. Scoop 32 tablespoon-sized balls of dough onto prepared sheets, spacing 1 1/2 inches apart.
7. Bake cookies one sheet at a time for 8-9 minutes, until edges are set and centers are puffed but soft.
8. Gently flatten cookies with the bottom of a measuring cup while still warm. Cool 10 minutes on the sheet, then transfer to a wire rack to cool completely.
9. For the filling, in a medium bowl, beat remaining 1/3 cup brown butter and cream cheese until smooth. Add powdered sugar, remaining 1 tsp vanilla, and 1/4 tsp salt. Beat until creamy.
10. Transfer filling to a piping bag or zip-top bag with a 1/2-inch opening.
11. Pair up cookies by size. Pipe filling onto the flat side of half the cookies. Top with remaining cookies, pressing gently to sandwich.