Add the butter to a skillet set over medium heat. Melt the butter and cook until the milk solids caramelize and the butter turns golden brown, approx. 5-8 minutes.
Transfer the butter to a heat-proof glass bowl. Set aside to cool to room temperature or refrigerate to cool the butter faster.
Chop the toffee bars into small pieces and set aside. Combine the flour, baking soda, baking powder, and salt in a mixing bowl and whisk together. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, combine the butter and sugars. Mix until light and creamy, approx. 3 minutes. Add the egg, egg yolk, and vanilla, mixing after each addition until fully incorporated.
Add the flour mixture and mix until just combined. Stir in the toffee and chocolate chips.
Using a 1-tablespoon cookie scoop, portion the dough onto a parchment-lined baking sheet. Press extra chocolate chips into the dough if desired. Chill for 30 minutes. (Note 1)
Preheat the oven to 375 degrees.
Transfer some of the cookie dough balls to another parchment-lined baking sheet, leaving about 2 inches between cookies. Bake the cookies for 8-9 minutes until lightly brown around the edges and just set in the middle. Sprinkle with flaky sea salt. Let cool for 3-5 minutes on the baking sheet before transferring them to a cooling rack. Store in an airtight container.
Don't skimp on the chilling time specified in a recipe. Chilling the dough chills the butter which prevents the cookies from over-spreading during baking. Chilling the dough also allows for the ingredients to meld and the flavors to enhance.
To freeze cookie dough, portion the dough using a cookie scoop and place it on a parchment-lined baking sheet. Put the pan in the freezer to flash-freeze the dough until firm. Then, transfer the dough balls to a freezer-safe bag and store them for up to 3 months.
To bake frozen dough: place the dough balls on a parchment-lined baking sheet and bake according to the recipe's directions, allowing an extra 1-2 minutes of baking time.
To freeze cookies: allow the cookies to cool completely. Place on a parchment-lined sheet pan in a single layer and freeze until firm. Transfer to a freezer-safe bag or container and store in the freezer for up to 1 month. To eat, let the cookies thaw at room temperature or warm briefly in the microwave.
For storing: to keep the cookies chewy, store a slice of bread in the container with the cookies. The moisture in the bread keeps the cookies soft. The bread will dry out and can be discarded.