Brown the butter. Melt the butter in a saucepan over medium heat, stirring frequently, until it becomes golden brown and smells nutty and fragrant. Pour into a glass mixing bowl.
Pour the 1/2 cup milk chocolate chips into the bowl of hot browned butter. Whisk until the chocolate has melted into the butter.
Mix the wet ingredients. Add the granulated sugar, eggs and egg yolk and whisk until smooth and well combined.
Add the room-temperature milk, vanilla extract, and sourdough discard. Whisk until the mixture is smooth and evenly combined.
Combine the dry ingredients. In a separate bowl, whisk together the all-purpose flour, Dutch-process cocoa powder, baking powder, salt, and espresso powder, if using.
Combine the batter. Add the dry ingredients to the wet ingredients and gently fold together just until no dry streaks remain. Be careful not to overmix.
Add the chocolate chips. Fold in the 1 cup of semi-sweet chocolate chips, reserving a handful to sprinkle over the top.
Bake. Transfer the batter to the prepared loaf pan and smooth the top. Sprinkle with the reserved chocolate chips. Bake at 350°F (175°C) for 60–70 minutes, or until a toothpick inserted into the center comes out without wet batter. A few moist crumbs are okay—the loaf should remain fudgy! It will bake more as it cools in the pan.
Allow the brownie bread to cool in the pan before removing it. For the cleanest slices, let it cool completely before cutting.