Melt butter in a medium saucepan (preferably a light color) over medium heat, stirring or whisking constantly until it foams, then browns, 5–8 minutes. Scrape into a large bowl. Let cool in the fridge for about 20 minutes, stirring occasionally.

In a bowl of an electric mixer or a large bowl using a hand mixer, add dark brown sugar and granulated sugar. Pour the brown butter over the sugar. Mix until light and fluffy, about 2 minutes. Add eggs and vanilla, increase mixer speed to medium-high, and beat until the mixture lightens and begins to thicken, about 1 minute. Reduce mixer speed to low; add baking soda and salt. Mix briefly, then add the flour and mix just until combined. Gently stir in the toffee pieces and chocolate chips.

Chill the dough for at least 2 hours or up to 48 hours. (Covered tightly with plastic wrap.) (You can either chill the dough and then scoop into balls or scoop into balls and then chill.) Scoop the dough into balls that are 2 or 3 tablespoons each.

Place a rack in the middle of the oven; preheat to 375°. Line a light-colored baking sheet with parchment paper.

If you didn’t already scoop the dough into balls, the dough will need some time to rest at room temperature, so it isn’t too hard to scoop the dough balls. Portion out balls of dough and place them on a parchment-lined baking sheet, spacing about 3" apart.

Bake the cookies until the edges are golden brown and firm, but the centers are still soft, 8-9 minutes. (If making cookies with 3 tablespoons of dough, bake for 9-10 minutes. Add more toffee pieces or chocolate chips if desired, and sprinkle a little flaky sea salt on each cookie. (Note: if cookies have spread a little, you can use the side of a fork to push them back in to make the cookies more circular.)

Let cool on baking sheets for 10 minutes, then transfer to a wire rack and let cool completely. Repeat with the remaining dough and a fresh parchment-lined, cooled baking sheet.

Store leftover cookies in an airtight container for around 3 days.