In a medium saucepan, brown the butter over medium-high heat. This takes me about 10 minutes. Your butter should melt, become frothy and foamy, and eventually will start to brown on the bottom. Transfer to a heat-safe glass measuring cup. Add enough water to the hot butter to make 1 1/2 cups total liquid in the measuring cup. Cover, and let the mixture cool on your counter to room temperature, about an hour.
Preheat your oven to 350 degrees, and line a 12 by 18 inch baking sheet with parchment paper. Set aside.
In a medium bowl, stir together the flour, cornstarch, baking soda, espresso powder, and salt. Set aside.
In a large bowl, beat together the browned butter mixture, brown sugar, and sugar until combined. Add in the eggs and vanilla, and beat until smooth. Add in the dry ingredients slowly, and beat until completely combined.
Scoop your dough onto your prepared baking sheet, and spread evenly. It should fill the entire sheet. Bake in your preheated oven for 15-17 minutes, or until the cookies have set around the edges and are just starting to become golden. Sprinkle the coarse salt over the cookies right after they come out of the oven. Cool in the sheet pan for 20 minutes before cutting and serving. Enjoy!