Preheat the oven to 350 degrees F. Butter an 8-inch square baking pan and dust it with the cocoa powder, tapping out the excess.
Fill a medium saucepan 1/3 full of water and bring it to a boil over medium-high heat. Put the butter, chocolate and espresso powder in a heatproof bowl, set it on the saucepan and turn off the heat. Let the bowl stand for about 5 minutes until the chocolate mixture is melted. Stir together until very smooth. Stir in the sugar until dissolved. Remove the bowl from the saucepan; let cool to room temperature.
Squeeze any excess moisture from the spinach and put it in a food processor. Add the vanilla and 1 tablespoon of water and puree until very smooth. Add the spinach puree and eggs to the cooled chocolate mixture and whisk until well combined.
Whisk together the flour, cocoa powder, baking powder and salt in a medium bowl until combined. Add the wet ingredients to the dry and fold with a rubber spatula until combined. Add the chocolate chips and stir to combine. Pour the batter into the prepared pan; use an offset spatula to spread the batter evenly. Bake until set and a toothpick inserted in the center still has a few moist crumbs stuck to it, about 25 minutes.
Cool the brownies completely in the pan before cutting.