Place butter in a saucepan over medium heat; cook, stirring constantly, until it turns amber and smells nutty. Remove from heat and let cool slightly.
In a large bowl, whisk the browned butter and brown sugar until smooth.
Add the egg, vanilla, and 1/3 cup of pumpkin butter, whisking until well combined.
Stir in the instant espresso powder until dissolved.
In a separate bowl, whisk together the flour, baking soda, salt, cinnamon, and ginger.
Gradually stir the dry ingredients into the wet mixture, then fold in the rolled oats.
Chill the dough for 30 minutes.
Preheat oven to 350°F (175°C) and line a baking sheet with parchment paper.
Scoop rounded tablespoons of dough onto the sheet. Make a small indentation in each and add a small dollop of the remaining pumpkin butter.
Bake for 10-12 minutes until edges are set.
Cool on the baking sheet for 5 minutes before transferring to a wire rack.