Prepare the baking sheet: Line a baking sheet with a silicone baking mat or parchment paper and set it aside.
Make the homemade toffee: Over medium heat, melt the butter then whisk in the brown sugar and vanilla for 1 minute. Cook the sugar for 8-10 mins, whisking occasionally, until a thermometer reads 295°F/146°C and it has a melted peanut butter consistency. If at any point the toffee begins to separate, remove the pan from the heat and whisk vigorously until it comes back together. Then pour the toffee onto the prepared baking sheet. Set aside to cool.
Smash the toffee into pieces: Once cool, (about the time it takes to brown and cool the butter for the cookie dough) use a meat mallet or heavy object to smash the toffee into pieces.
Brown the butter: Cook the butter in a stainless steel pan over medium heat until it looks foamy, smells nutty, and has browned bits of milk solids. The butter will foam, crackle, and pop as it cooks- this is normal! Scrape into a large bowl and cool to room temperature (20-30 minutes). You should have about ¾ cup (160 grams) of browned butter.
Whisk in the wet ingredients: Once cool, whisk in the brown sugar and granulated sugar. Then whisk in the egg, egg yolk, and vanilla until combined.
Mix in the dry ingredients: Fold in the flour, baking soda, salt, chocolate chips, and toffee pieces until just combined. Cover the bowl with plastic wrap and chill it in the fridge for 30 minutes, this makes it easier to scoop!
Scoop and chill the cookie dough: Scoop the dough with a 3-tbsp scoop onto a parchment-lined baking sheet. Chill the tray in the freezer for a minimum of 3-4 hours, overnight is best! They must be frozen when baking!
Bake the cookies: When ready to bake, preheat the oven to 350°F/180°C. Space the cookies 2-3 inches apart. You will need to bake them on two trays if baking the whole batch! Bake for 12-16 minutes, or until the edges are golden brown but the middle is puffy and underbaked.
Straight out of the oven, sprinkle with extra toffee pieces, chocolate chips, and flaky sea salt if desired. Let cool completely on a wire rack then enjoy!