Preheat oven to 350°F. Line an 8x8 square pan with foil or parchment paper, leaving a couple inches of overhang on two sides (this will allow you to lift the whole tray of brownies out of the pan for easy removal). Grease the foil/parchment.
Combine granulated sugar and melted butter in a large bowl. Stir it briefly then allow it to sit undisturbed for 5 minutes. The heat from the butter will begin to dissolve the sugar. After 5 minutes, whisk the butter and sugar vigorously for 30 seconds. Add eggs, water, and vanilla extract, whisking again for about 30 seconds. Tap all the excess batter off the whisk and remove the whisk from the bowl.
Sprinkle cocoa powder, powdered sugar, flour, chocolate chips, and salt over the wet ingredients. Use a silicone spatula or wooden spoon to stir the mixture until it is glossy and well combined.
Pour the batter into the prepared pan and smooth the top. Bake for for 28-33 minutes (begin checking at 28 minutes). The brownies are done when a toothpick inserted into the center comes out without raw batter on it, moist fudgy crumbs are OK. Be careful not to over bake. Remove from the oven and place on a cooling rack to cool completely.
Cookie dough frosting:
Using a hand mixer or stand mixer fitted with the paddle attachment, beat butter, brown sugar, granulated sugar, and salt until smooth and combined. Add vanilla and beat until combined, followed by flour. Add the milk, one tablespoon at a time, until you reach a thick but spreadable consistency (I used 3-tablespoons of milk). Stir in the chocolate chips.
Scoop the cookie dough mixture on top of the brownies and spread into an even layer. Place in the refrigerator for an hour or two for the topping to firm up.
Notes
Cookie dough topping (optional step): The topping for these brownies is a raw cookie dough. The cookie dough is eggless, so you don't have to worry about bacteria from raw eggs, but there is a minimal risk related to eating raw flour. If you are concerned about this you can heat treat the flour for the cookie dough topping. Place the flour in a microwave safe bowl and heat on high for 1 minute. Cool and then proceed with cookie dough topping. Note: you do not need to do this for the flour in the brownies because it will be cooked in the oven.Serving and storage: Serve chilled or at cool room temperature. Store brownies in the refrigerator for up to 3 days. If you have any brownies leftover you can freeze them (double wrapped) for up to 3 months. Thaw in the refrigerator.