Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper.
Cream butter and 1 cup sugar until light and fluffy.
Add egg and vanilla extract, mixing until combined.
In a separate bowl, whisk flour, baking soda, cream of tartar, and salt. Add to the wet ingredients and mix until a soft dough forms.
Combine 1/4 cup sugar and cinnamon in a small bowl.
Scoop approximately 1 1/2 tablespoons of dough and flatten gently.
Place one caramel candy in the center and wrap the dough completely around it. Roll into a smooth ball.
Roll the stuffed dough ball in cinnamon sugar.
Arrange cookies 2 inches apart on the prepared baking sheet.
Bake for 10–12 minutes until puffed and lightly golden around the edges.
Cool on the baking sheet for 5 minutes, then transfer to a wire rack.