Melt the butter in a saucepan over medium heat, cooking until it turns golden brown and gives off a nutty aroma. Remove from heat and let cool slightly.
In a bowl, combine the browned butter with brown sugar and granulated sugar, mixing until smooth.
Add the beaten egg and vanilla extract to the mixture and stir until fully incorporated.
In a separate bowl, whisk together the flour, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Fold in the crushed toffee bits evenly through the dough.
Preheat the oven to 350°F (177°C).
Drop rounded tablespoons of dough onto a lined baking sheet, spacing them 2 inches apart.
Bake cookies for 10-12 minutes or until edges are golden brown. Remove and cool on wire racks.
Notes
For extra toffee flavor, sprinkle additional toffee bits on top before baking.