Brown the butter: Add butter to a light-colored medium pot and melt over medium heat. Using a rubber spatula, stir constantly as the butter melts and begins to foam. As the butter begins to foam, add milk powder and continue stirring until nutty in aroma and brown specks form at the bottom of the pan. Once golden brown, remove from heat and pour brown butter into a heat-proof bowl. Allow to cool for 5-10 minutes.
In a small bowl, combine all-purpose flour, baking soda, salt, and espresso powder, and whisk to combine. Set aside.
Add light brown sugar and granulated sugar to a large bowl. Add slightly cooled brown butter and whisk to combine. Then, add egg, egg yolk, and vanilla bean paste and continue whisking until incorporated and slightly lightened in color.
Add dry ingredients and mix just until incorporated. Switch to a wooden spoon or rigid spatula, add toffee and chopped chocolate (except for reserve chocolate), and stir together until evenly distributed.
Chill dough in the refrigerator for at least 1 hour, or up to overnight. Towards the end of your chill time, preheat the oven to 375°F.
Scoop dough into 1-ounce portions and place on a baking sheet lined with parchment paper. No need to roll into perfect balls. For a half sheet pan, you should be able to fit 6 cookies, arranged 2-1-2-1. Press reserve chocolate pieces into the top of the dough balls (this gives you those beautiful pools of melted chocolate when they come out of the oven).
Bake until golden brown around the outside and still soft in the middle, ~12 minutes. If desired, immediately after removing, use a cookie cutter or bowl to round any misshapen cookies into perfect circles.
Let cool in the pan for 15 minutes before transferring to a cooling rack to cool completely.
Sprinkle generously with flaky sea salt to finish.