Adjust rack to lower third of oven and preheat oven to 350 degrees F. Using an electric mixer on medium-low speed, cream butter in a large bowl until smooth. Increase mixer speed to medium, add sugar, and mix until well combined and slightly fluffy. Lower mixer speed and add egg white and vanilla, then the flour in two additions, scraping down the sides and mixing until thoroughly combined.
Using 2 level teaspoonfuls of dough, form a flat disk in the palm of your hand. Center a caramel half on the dough and bring the dough up to cover it completely. Roll gently in hands to form a ball. Place balls 1 inch apart onto parchment-lined baking sheets. Bake for 8 to 10 minutes, or until cookies are set and no longer shiny but dull, and the bottoms are browned. Place baking sheets onto a wire rack and allow to cool completely before lifting cookies from parchment paper.