Preheat the oven to 350°F/175°C and line two large baking sheets with parchment paper or silicone baking mats. Set aside.
Add 1/2 cup of room-temperature brown butter (the remaining amount will be used to make the frosting) and 1 cup packed dark brown sugar into a large bowl or the bowl of a stand mixer fitted with a paddle attachment. Beat on a medium-high speed for a couple of minutes, or until the mixture becomes lighter in color and looks fluffy. Scrape the sides and bottom of the bowl as needed with a rubber spatula.
Next, scoop 1/4 cup of pumpkin puree onto a couple of sheets of paper towel and pat it to remove any excess moisture. Then add the blotted pumpkin puree (or 1 large egg) and 1 tsp vanilla extract to the butter/brown sugar mixture. Mix on a medium speed until combined.
Then fold 1 1/4 cups all-purpose flour, 3/4 cup old-fashioned oats, 1 tsp baking soda, 1/2 tsp ground cinnamon, and 1/2 tsp fine salt into the wet ingredients with a rubber spatula or wooden spoon until just combined. Scrape the sides and bottom of the bowl as needed to make sure everything is properly mixed together. The dough should be quite thick.
Use a 1 1/2 Tbsp cookie scoop or a small spoon to measure about 20 cookie dough balls and place them on the prepared pans. Space them about 1 1/2 inches apart, as they'll spread a bit as they bake.
Bake one tray at a time on the middle rack of your oven for 9-10 minutes, or until the edges look set and the centers have puffed up but are still soft.
This is optional, but I like to gently flatten the cookies with the bottom of a metal measuring cup or glass while they're still warm to make them even chewier. Then let the cookies cool on the sheet for 10 minutes, then place them on a wire rack to finish cooling.