In a large bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy.
Beat in the egg and vanilla extract until well combined.
In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add the dry ingredients to the wet mixture, mixing until just incorporated.
Fold in the chopped toffee bits until evenly distributed throughout the dough.
Divide the dough into two equal portions and shape each into a 12-inch log about 2 inches wide. Wrap the logs tightly in plastic wrap and refrigerate for at least 4 hours, or up to 3 days.
Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
Slice the chilled dough logs into 0.5-inch thick rounds and place them on the prepared baking sheets, spacing them about 2 inches apart.
Bake for 10–12 minutes, or until the edges are lightly golden. Cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
Notes
Store wrapped logs in the refrigerator for up to 3 days or freeze for longer storage.