Preheat the oven to 350° F. Line a baking sheet with parchment paper.Add the butter to a skillet set over medium heat. Cook until the butter begins to brown, about 3-4 minutes. Remove from the heat and transfer to a heatproof bowl. Let cool 5 minutes or so.To the browned butter, mix in the brown sugar, 1/3 cup pumpkin butter, the egg, vanilla, and espresso powder, mixing until smooth. Add the flour, oats, baking soda, pumpkin spice, and salt.Dollop the remaining 1/3 cup of pumpkin butter over the dough. Gently fold the butter into the dough, being careful to not over-mix. You want streaks of pumpkin butter.Roll the dough into rounded tablespoon-size balls, it might be a little sticky. Place 2 inches apart on the prepared baking sheet. Gently flatten the dough down. Bake 8 minutes. Remove from the oven, rotate, and tap the baking sheet on the counter 1-2 times to flatten. Bake another 2-3 minutes or until the cookies are just beginning to set on the edges.Let the cookies cool on the baking sheet. They will continue to cook slightly as they sit on the baking sheet.Meanwhile, make the icing. Add the butter to a pot set over medium heat. Allow the butter to brown lightly until it smells toasted, about 2-3 minutes. Remove from the heat. Whisk in the powdered sugar, vanilla, a pinch of salt. Immediately spread the icing over the cookies moving fast, as it will set quickly. Store in an airtight container for up to 5 days.