Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a mixing bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, and nutmeg. Set aside.
In another bowl, beat together the softened butter, brown sugar, and granulated sugar until creamy. Add in the pumpkin puree, vanilla extract, and egg, mixing until well combined.
Gradually add the dry ingredients to the wet mixture, stirring until just combined. Do not overmix.
For the crumble, mix the oats, flour, brown sugar, and cinnamon in a small bowl. Pour in the melted butter and stir until crumbly.
Drop tablespoon-sized amounts of cookie dough onto the prepared baking sheet, spacing them about 2 inches apart. Sprinkle the crumble topping generously over each cookie.
Bake in the preheated oven for 10-12 minutes or until the edges are lightly golden. The centers will still be soft.
Let the cookies cool on the baking sheet for 5 minutes before transferring them to a cooling rack to cool completely.
For extra flavor, consider adding chocolate chips or nuts to the cookie dough.
Store leftover cookies in an airtight container at room temperature for up to 5 days.