Sift together the cake flour, bread flour, baking soda, baking powder, and kosher salt into a bowl. Set aside.
In a separate bowl, using a mixer fitted with a paddle attachment, combine the cooled brown butter and both sugars, mixing at medium speed for about three minutes. You should see the mixture become light and fluffy, and the delightful aroma of the brown butter will fill your kitchen.
Add the eggs and the additional egg yolk, one at a time, mixing well after each addition. This ensures that the eggs are fully incorporated, contributing to the cookies' texture and richness.
Stir in the vanilla extract, letting the warm, comforting flavor meld with the butter and sugars. This is where the magic starts to happen. Mix until everything is well combined.
Reduce the mixer speed to low and gradually add the sifted dry ingredients, mixing until just combined. Be careful not to overmix; you want a soft dough. This step should only take about 5 to 10 seconds.
Now it’s time to add the dark chocolate chips and Heath Milk Chocolate English Toffee Bits. Gently mix until they’re evenly distributed throughout the dough, creating a tempting cookie mixture.
To ensure the best texture, press plastic wrap against the dough and refrigerate for 24 to 36 hours. This resting period allows the flavors to develop and the dough to firm up, resulting in cookies that are truly delightful.
When you're ready to bake, remove the dough from the fridge and let it sit at room temperature for about 30 minutes to 1 hour. This will make it easier to scoop.
Preheat your oven to 350°F. Line a baking sheet with parchment paper, ensuring that the cookies won’t stick.
Scoop dough balls of approximately 3 ounces each and place them onto the prepared baking sheet, leaving a few inches between each cookie for spreading. The anticipation will build as you watch the dough transform in the oven!
Bake in the preheated oven for 16 to 18 minutes, or until they are golden brown but still soft in the center. If you want to take it up a notch, sprinkle some flaky sea salt on top right after they come out of the oven.
Transfer the baking sheet to a wire rack for about 10 minutes, allowing the cookies to set before moving them to another rack to cool completely. Repeat this process with any remaining dough, or save it for later baking!