In a mixer fitted with a paddle attachment, cream butter and sugar until light and fluffy.
Add egg and vanilla extract. Mix until combined.
In a medium bowl, whisk together flour, salt, baking powder, baking soda, and cream of tartar. Add to mixer on a slow speed. Mix until just combined.
Stir in Butterfinger Bits.
Using a medium scooper, scoop out dough balls and put them in the fridge for at least an hour.
Preheat the oven to 350 degrees.
Place the chilled cookies on a baking sheet with parchment paper and bake for 8-10 minutes.
Let cookies sit for 1 - 2 minutes and then add Reese's Peanut Butter Cup pieces on top.
Allow cookies to cool on baking sheet.