Melt the butter in a sauce pan over medium-high heat, stirring often, until the butter begins to foam and then brown. Remove from heat and pour into the bowl of an electric mixer fitted with the paddle attachment.
Add the brown sugar and granulated sugar to the brown butter and mix on low for 30 seconds until just combined. Let the mixture cool slightly, until lukewarm.
In a small bowl, sift together the flour, baking soda, and salt
Add the eggs and vanilla to the butter-sugar mixture and mix on medium speed until batter begins to thicken and lighten in color, about 1-2 minutes. Scrap the bowl down to ensure mixture is well combined.
With the mixer on low, slowly add flour mixture to the butter mixture. Mix until just combined.
With a rubber spatula, fold in the toffee bits and chocolate chips. Set the batter aside and let the flour hydrate for at least 30 minutes. The batter will be very loose, but will thicken as it sits.
Preheat your oven to 375° or 350° if using a convection oven.
If making smaller cookies, use a tablespoon to portion out the dough and place 1-2 inches apart on a parchment lined baking sheet. If making larger cookies, use an ice cream scoops to portion out dough and place 3 inches apart.
Bake cookies for 9-11 minutes, or until edges are just beginning to brown and crisp slightly. Remove from oven and let cool for 10 minutes. Transfer cookies to a wire rack and cool completely.
Repeat with remaining dough, or store left over cookie dough in refrigerator for about 4 days or in freezer for 2 weeks.