Preheat the oven to 350˚F (180˚C). Line a baking sheet with parchment paper.
Make the cookie butter: In a food processor, combine the butter, evaporated milk, sweetened condensed milk, and cookies. Process until the cookies break down and the mixture is the texture of peanut butter.
Using a small ice cream scoop or your hands, scoop 12 ½-inch (1 ¼ cm) balls of the cookie butter onto the prepared baking sheet. Freeze for 30 minutes.
Make the cookie dough: In a large bowl, whisk together the butter, granulated sugar, brown sugar, and vanilla until light and fluffy.
Add the egg and whisk to incorporate.
Sift in the flour, cinnamon, cream of tartar, baking soda, and salt. Fold with a rubber spatula to combine.
Cover the bowl with plastic wrap and chill the dough for 30 minutes.
Make the cinnamon sugar: In a small bowl, mix together the sugar and cinnamon. Set aside.
Roll the cookie dough into 24 ½-inch (1 ¼ cm) balls.
Flatten a dough ball in your hand and add a cookie butter ball to the center. Top with another flattened ball of cookie dough and pinch the edges of the dough together. Roll into a ball to completely encase the cookie butter. Repeat with the remaining ingredients.
Roll the cookie dough balls in the cinnamon sugar to coat.
Place the cookie balls on a parchment-lined baking sheet, spacing about 3 inches (7 ½ cm) apart – there should be 6 per baking sheet.
Bake for 12 minutes, until the edges are light golden brown.