Preheat oven to 350°F (175°C). Line two baking sheets with silicone baking mats or parchment paper and set aside.
In a large bowl, cream together the softened butter, natural chunky peanut butter, packed brown sugar, and granulated sugar until light and fluffy.
Add the egg and vanilla; mix until incorporated, scraping down the sides of the bowl as needed.
In a separate bowl, whisk the all-purpose flour and baking soda together to combine.
Gradually add the flour mixture to the wet ingredients and mix just until combined; avoid overmixing.
Fold the toffee bits into the dough by hand until evenly distributed.
Scoop dough onto the prepared baking sheets (about 8 dough balls per sheet), leaving plenty of space between each for spreading.
Press each dough ball lightly with a fork to flatten slightly.
Bake in the preheated oven for 7–8 minutes, until the edges are set and centers are just beginning to firm.
Cool the cookies on the baking sheet for 3 minutes, then transfer to a wire rack to cool completely. Store in an airtight container for up to 3 days.