Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper.
Melt the butter in a light-colored pan over medium-low heat, stirring occasionally until it turns golden brown and smells nutty. Pour the browned butter into a bowl and let it cool until just warm.
Add the sugar to the butter and mix well.
Add the eggs, one at a time, whisking to combine after each addition. Then, add the pumpkin puree and vanilla, whisking until the mixture is smooth and homogeneous.
Sift the flour, baking powder, salt, and cinnamon into the wet mixture. Mix everything together just until no dry ingredients remain.
Using a 3-tablespoon cookie scoop (or a similar-sized scoop), place scoops of dough onto the prepared baking sheet, leaving 1 ½ inches (4cm) between each. You will make approximately 28 cookies (14 whoopie pies), so you may need to bake in 2-3 batches. The dough can sit at room temperature while each batch bakes.
Bake for 12-15 minutes, or until the cookies have spread, puffed up, and are springy to the touch.
Remove from the oven and let the cookies cool completely before assembling.