Place butter in a light-colored saucepan over medium heat. Stir constantly as it melts, foams, and then quiets down. Continue cooking until the milk solids turn deep amber and the butter smells nutty, about 5-7 minutes. Pour immediately into a large mixing bowl and let cool for 10-15 minutes.

The butter will be liquid. That's correct. Do not refrigerate it to speed up cooling - you want it fluid but not hot.

Add dark brown sugar and granulated sugar to the cooled brown butter. Whisk until combined and slightly glossy, about 1 minute.

Add the whole egg, egg yolk, and vanilla extract. Whisk vigorously for about 90 seconds until the mixture looks pale and thick.

Add flour, baking soda, and fine sea salt. Fold with a rubber spatula until just combined and no dry streaks remain.

Fold in the toffee bits until evenly distributed throughout the dough.

Cover the bowl and refrigerate the dough for at least 30 minutes. The dough will firm up and become scoopable.

Heat the oven to 175 C / 350 F. Line two baking sheets with parchment paper.

Scoop dough into balls about 2 tablespoons each (roughly 40 g). Place them 5 cm apart on the prepared pans. Press a few extra toffee bits onto the top of each ball. Add a pinch of flaky salt if using.

Bake one sheet at a time on the center rack for 10-12 minutes, until the edges are set and golden but the centers still look slightly underdone.

Remove the pan from the oven. Tap the pan firmly on the counter once to help the cookies settle. Let cookies cool on the pan for 5 minutes before transferring to a wire rack. They will firm up as they cool.