In a large bowl, beat together the butter, dark brown sugar, and granulated sugar until combined and fluffy. Add the peanut butter, vanilla, and egg and mix until well combined, about 1 minute.
In a medium bowl, whisk together the baking soda, salt, and flour. Add the dry ingredients to the wet ingredients and mix until just combined; be sure not to overmix! Fold in ¾ cup (105 grams) of the Butterfinger Bits until incorporated.
Using a large (3-tablespoon) spring-loaded cookie scoop, scoop the dough and roll it into balls, then place on a parchment paper-lined baking sheet. Loosely cover with plastic wrap, and place in the refrigerator to chill for at least 30 minutes.
Preheat the oven to 350°F (180°C). Line another baking sheet with parchment paper and set aside.
Bake the cookies for 9-10 minutes until lightly browned on the sides; the centers will look very soft. Remove from the oven and allow to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
While the cookies are cooling, add the chopped chocolate to a large heatproof bowl, then set the bowl over a saucepan filled with just two inches of simmering water over medium heat. Do not let the bottom of the bowl touch the water. Stir until the chocolate is completely melted, then remove from the heat and set aside to cool slightly.
Dip half of each cookie into the melted chocolate, scrape the bottom of the cookie on the bowl of the melted chocolate, and transfer to the empty baking sheet lined with parchment paper. Sprinkle with remaining Butterfinger bits and allow chocolate to set at room temperature.
These cookies will remain fresh in an airtight container at room temperature for 4-5 days.