First, make your brown butter. In a small saucepan, melt ONE (1) stick of butter over medium-low heat. The butter will eventually melt, then start to bubble, pop and foam up, turning frothy. This is normal. Cook low and slow for about 5-7 minutes, swirling the butter around the pan occasionally, until the butter is amber in color and fragrant, almost nutty. Do NOT burn the butter! Once butter has browned and is amber and fragrant, remove from heat and pour into a heat-proof bowl to cool, about 20 minutes.
In the bowl of your stand mixer, cream together the butter, brown sugar and white sugar with the paddle attachment for approx. 2 minutes, until creamy and fluffy. Add in the egg and vanilla bean paste to combine. Add in the salt, espresso powder, baking soda and cornstarch, and beat to combine. Add in HALF of the flour, beating to combine. Stream in HALF of the browned butter. Add the remaining flour, beating to combine, then stream in the remaining browned butter, adding in all the bits at the bottom. Once dough comes together, stir in the chocolate chunks.
Refrigerate the dough for AT LEAST 2 hours, but I recommend refrigerating for 4 hours to overnight. This step is MANDATORY as the dough will spread significantly if it does not sufficiently chill. The dough needs to be super duper cold before baking, so I really stress chilling for at least four hours.
Preheat oven to 350 degrees F. Line 2 baking sheets with silicone liners or parchment sprayed with cooking spray. Drop 1/4-cup rounded mounds of cookie dough onto the baking sheets, keeping about 2-3" apart. Sprinkle the tops with sea salt as desired.
Bake for approx. 10-13 minutes, rotating pans halfway through baking time to ensure even cooking. Cookies may appear slightly under-done; this is normal and they will continue to firm up as they cool. If desired, add a couple more chocolate chunks on top of the freshly baked cookies for a prettier appearance. Allow the cookies to cool completely on the baking sheets before serving. Store leftovers airtight at room temperature for up to a week.