Brown the butter: In a saucepan over low heat, cook the butter until small brown flakes form, stirring or swirling frequently to avoid burning. The butter will foam and bubble, but keep stirring and checking the color. Once the butter turns amber/brown and smells nutty (about 7-10 minutes), pour the melted butter and browned bits into a heatproof bowl. Let it cool and solidify back to room temperature (about 1 hour).
Prepare dry ingredients: In a medium bowl, whisk together the flour, baking soda, cornstarch, espresso powder, and kosher salt. Set aside.
Cream the butter and sugars: In a stand mixer fitted with a paddle attachment, or using a hand mixer, beat the browned butter, dark brown sugar, and granulated sugar on medium-high speed until light and fluffy (about 4-5 minutes).
Add eggs and vanilla: Scrape down the sides of the bowl, add one egg and vanilla extract, and mix until well combined (30-60 seconds). Scrape the bowl, add the second egg, and mix again until fully incorporated.
Combine wet and dry ingredients: With the mixer on low, gradually add the flour mixture in three batches, scraping down the sides of the bowl before adding the last portion. Mix until just a few streaks of flour remain.
Fold in the toffee bits: Gently fold in the toffee bits using a spatula.
Chill the dough: Using a size 30 scoop (a little over 2 tablespoons), portion the dough into 24 equal scoops. Chill the dough for 1 hour.
Bake: Preheat the oven to 350°F (175°C) and line baking sheets with parchment paper or a silicone baking mat. Place the chilled cookie dough scoops on the prepared baking sheets, about 2 inches apart. Bake for 8-10 minutes, until the edges are lightly browned and crisped. The centers may look slightly underdone.
Create the crinkle effect: Once removed from the oven, drop the baking pan hard on the counter once to create crinkled edges and deflate the cookies slightly. If the cookies are puffier than desired, gently press them down with the flat bottom of a cup. Sprinkle with flaky sea salt, if desired.
Cool and serve: Allow the cookies to cool on the pan for 2 minutes, then transfer them to a wire rack to cool completely.