Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper or a silicone baking mat.
In a large mixing bowl, use an electric mixer to cream together the softened butter and granulated sugar until light and fluffy, about 3 to 5 minutes.
Beat in the egg and vanilla extract, mixing until fully incorporated and the dough is smooth and glossy.
In a separate bowl, whisk together the all-purpose flour, baking powder, ground cinnamon, and salt to evenly distribute the spices and leavening agents.
Gradually add the dry ingredients to the wet mixture, stirring on low speed or folding gently with a spatula until just combined. Avoid overmixing.
Scoop dough into 1-inch balls and place them 2 inches apart on the prepared baking sheet. Using your thumb or the back of a teaspoon, gently press an indentation into the center of each ball to create the thumbprint.
Spoon about 1/2 teaspoon of apple butter into each indentation, being careful not to overfill.
Bake the cookies for 12-15 minutes, or until the edges are lightly golden and the centers look set. The apple butter should be bubbly but not burnt.
Allow the cookies to cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely.