Preheat the oven to 350F [180C] and set two racks at the upper-middle and lower-middle positions. Line two baking sheets with parchment paper. In a medium heatproof bowl, add the chocolate chips, warm water, and vanilla. In a small saucepan over medium heat, melt the butter. Continue cooking, stirring often, until the butter foams and then darkens in color slightly and is fragrant, 3 to 4 minutes.

Slowly pour the hot butter over the chocolate chips. (Be careful – the mixture will sizzle.) Whisk until the chocolate is melted and the mixture is completely smooth. Set aside.

Add the eggs, granulated sugar, and brown sugar to the bowl of a stand mixer fitted with the whisk attachment. (Alternatively, use a hand mixer and large bowl.) Whip on high speed until the eggs are pale yellow, ribbony, and almost tripled in volume, about 6 minutes. As the eggs are whipping, sift together the flour, cocoa powder, baking powder, espresso powder, and salt in a medium bowl. Reduce the mixer speed to low and slowly pour the melted chocolate mixture into the eggs.

Sprinkle the tops with flaky sea salt (if using). Bake both sheets at the same time, swapping them midway, until the tops of the cookies are cracked and shiny but they still appear slightly underdone, 9 to 11 minutes. Let cool slightly on the baking sheets, then transfer to a wire rack to cool completely. Cookies will keep in an airtight container in a single layer for up to 1 week.