Line baking sheets with parchment paper and set aside.
Add the cut-up butter to a large, light-colored skillet or pot. Melt the butter over medium heat, whisking constantly. After the butter melts it will start to foam, but continue whisking it for another 5 to 7 minutes, until the butter begins browning and there are lightly browned pieces on the bottom of the pan. The butter will have a nutty aroma. Once browned, remove the pan from the heat and immediately pour the butter into a large, heatproof bowl. Let the butter cool for 5 minutes.
Add the brown sugar and granulated sugar to the butter and mix until smooth. Mix in the vanilla extract and eggs until well combined.
In a separate bowl, whisk together the flour, baking soda, and salt. Mix the dry ingredients into the butter mixture until completely combined. Fold in the chocolate toffee bits until they are well incorporated.
Use a medium-sized cookie scoop or spoon to drop rounded scoops onto the prepared cookie sheets, spacing them at least 2 inches apart. Chill in the refrigerator for 15 to 30 minutes.
Preheat the oven to 350 F. Bake the chilled cookie dough in the preheated oven for 12 to 14 minutes, until the edges begin to turn golden brown. Allow cookies to cool for 5 minutes, then transfer them to a cooling rack to cool completely.