In a small bowl, mix together the granulated sugar, cinnamon, and apple pie spice. Set aside.
In a medium bowl, whisk together the flour, apple pie spice, baking powder, and salt.
In a large bowl, beat the softened butter and granulated sugar until light and creamy, about 2 minutes. Add the egg and vanilla and mix until combined. Gradually add the dry ingredients, mixing on low just until a soft dough forms. Do not overmix.
Scoop the dough into 1-inch balls, about 1 tablespoon each. Roll each ball in the cinnamon-sugar mixture, coating it evenly. Place the cookies on a parchment-lined baking sheet.
Use your thumb or the end of a ½-teaspoon measuring spoon to make a deep indent in the center of each cookie. If the dough cracks around the edges, gently press it back together.
Cover and chill the shaped cookies for at least 2 hours, or up to overnight. Chilling helps the cookies hold their shape while baking.
When ready to bake, preheat the oven to 350°F. Arrange the chilled cookies on parchment-lined baking sheets, spacing them about 2 inches apart. Bake for 11 to 12 minutes, or until the edges are set and the centers still look slightly soft.
Remove the cookies from the oven. If the centers have puffed up, gently press the indents down again using the end of a spoon or measuring spoon. Spoon about ½ teaspoon of apple butter into the center of each cookie.
Return the cookies to the oven and bake for another 1 to 2 minutes, just until the apple butter is warmed through and slightly set.
Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely. The apple butter will set a little more as the cookies cool.