In a large mixing bowl, beat the softened butter and 3/4 cup of granulated sugar together until the mixture is light and fluffy. This should take about 2 to 3 minutes using an electric mixer.

Add the egg, apple butter, and vanilla extract to the butter mixture. Continue to beat on medium speed until everything is well combined and smooth.

In a separate medium bowl, whisk together the all-purpose flour, cream of tartar, baking soda, and salt. Gradually add these dry ingredients to the wet ingredients, mixing on low speed just until the flour disappears. Do not overmix.

Cover the bowl with plastic wrap and place it in the refrigerator for at least 30 minutes. Chilling the dough makes it easier to handle and prevents the cookies from spreading too much in the oven.

Preheat your oven to 350°F (175°C) and line two large baking sheets with parchment paper. In a small shallow bowl, stir together the 1/4 cup of sugar and 1 tablespoon of cinnamon for the coating.

Scoop about one tablespoon of dough at a time and roll it into a 1-inch ball between your palms. Roll each ball in the cinnamon-sugar mixture until it is completely coated.

Place the balls on the prepared baking sheets, spaced about 2 inches apart. Bake for 10 to 12 minutes, or until the edges are set but the centers still look slightly soft. Let them cool on the pan for 5 minutes before transferring to a wire rack.