Start by preheating your oven to 350°F (175°C). Line two large baking sheets with parchment paper or silicone baking mats to prevent sticking and ensure even browning.

In a large mixing bowl, beat the softened butter, granulated sugar, and light brown sugar together using a hand mixer or stand mixer. Continue mixing on medium-high speed for about 2-3 minutes until the mixture is light, fluffy, and pale in color.

Add the egg, apple butter, and vanilla extract to the butter mixture. Beat well until fully combined. The batter may look slightly curdled due to the moisture in the apple butter, but it will smooth out once the dry ingredients are added.

In a separate medium bowl, whisk together the flour, cream of tartar, baking soda, salt, and cinnamon. Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix the dough.

Scoop about 1.5 tablespoons of dough and roll it into a ball between your palms. Roll each ball in the prepared cinnamon-sugar coating until evenly covered. Place the balls on the baking sheets, spaced about 2 inches apart.

Bake for 9 to 11 minutes, or until the edges are set and the tops look slightly cracked but still soft. Let the cookies rest on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.