For the cookies: Whisk the flour, baking powder, salt and 1/2 teaspoon of the pumpkin pie spice in a medium bowl. Beat the butter and 1/2 cup of the granulated sugar in a stand mixer fitted with the paddle attachment on medium-high speed until light and fluffy, about 3 minutes. Beat in the egg until combined. Add the vanilla. Reduce the mixer speed to low and beat in the flour mixture until the dough comes together.
Mix the remaining 1/4 teaspoon pumpkin pie spice and 1/4 cup granulated sugar in a shallow bowl or plate. Roll the dough into 1 1/2-inch balls (each ball should be a heaping tablespoon; you should have about 18 balls), then roll in the spiced sugar mixture. Divide the dough balls between 2 baking sheets, spacing them about 2 inches apart. Use the handle of a wooden spoon to indent the centers of each dough ball, pinching together any large cracks that form (a few small cracks are fine). Refrigerate until firm, 20 to 30 minutes.
Position oven racks in the upper and lower thirds of the oven and preheat to 375 degrees F.
Bake until the cookies just begin to warm through, about 6 minutes. Remove the baking sheets from the oven and use the handle of a wooden spoon to indent the centers just a little more. Return them to the oven, rotating the baking sheets from top to bottom and front to back, and bake until the cookies are set and slightly browned on the bottom, an additional 10 to 12 minutes. Transfer the cookies to a wire rack to cool completely, about 30 minutes.
For the pecan crumble: Meanwhile, microwave the butter in a small microwave-safe bowl until melted, about 1 minute. Stir in the pecans, brown sugar, oats and pumpkin pie spice. Microwave until the sugar is dissolved, about 1 minute. Let cool in the bowl until the crumble is crunchy, about 30 minutes.
For the pumpkin pie filling: Whisk the confectioners’ sugar, pumpkin puree, cream cheese, vanilla extract and fine salt in a small bowl until smooth.
Assemble the cookies: Once the cookies are cool, spoon 1 to 2 teaspoons of filling into the center of each cookie and smooth the top if needed. Sprinkle the crumble on top of the filling. Store in an airtight container for up to 3 days.