Cut the butter into cubes. Take half of the butter (½ cup or 133g) and place it in a large saucepan. Set the pan over medium heat and stir until melted.
When the butter begins to bubble, add in the nonfat dry milk powder (if using). Continue to stir the butter every ten seconds or so, until it bubbles and begins to foam. Once you see brown bits floating in the butter and a nutty aroma is released, immediately remove the pan from the heat.
Pour the butter into a bowl to stop the cooking process, making sure to get all the browned bits from the bottom of the pan. Immediately add in the other half of the cubed butter, whisking until completely melted. At this point, the butter should be cooled to room temperature.
Add in the brown sugar and granulated sugar and whisk for 1-2 minutes until well combined, then beat in the eggs, egg yolk, and vanilla extract.
Once the mixture is smooth, fold in the flour, espresso powder, baking soda, and salt. Just before the flour is fully combined, add in the chopped chocolate and walnuts/toffee (if using) and mix until there are no more streaks of flour left. Do not overmix.
Scoop the dough onto a baking sheet lined with wax or parchment paper (no need to space them apart). Cover the tray with plastic wrap and refrigerate for at least 2 hours but up to 4-5 days. At this point, the cookie dough can be frozen.1 You can bake the cookies immediately, but they will spread more and have less flavor.
Preheat the oven to 375℉ and line a baking sheet with parchment paper.
When ready to bake, evenly space the cookie dough balls onto the baking sheet with a 2'' gap between each cookie. If desired, press the top of the dough with more chopped chocolate pieces.
Bake for 10-12 minutes, rotating the tray halfway through. Remove the cookies from the oven once the edges are golden brown and set, but do not overbake. The inside of the cookies should still be gooey and slightly raw, as they will continue to cook as they cool. Once out of the oven, sprinkle the cookies with flaky salt and allow to cool before serving.