Preheat oven to 350 degrees F. Spray an 8-inch square baking pan with cooking spray. Whisk ½ cup sugar with ¼ cup cocoa powder and set aside.
Whisk the flour, remaining ½ cup sugar, remaining ¼ cup cocoa, baking powder, and salt in a large bowl. Whisk milk, butter, egg yolk, and vanilla in medium bowl until smooth. Stir milk mixture into flour mixture until just combined. Fold in chocolate chips (batter will be stiff).
Using a rubber spatula, scrape batter into prepared pan and spread into corners. Sprinkle reserved cocoa and sugar mixture evenly over the top. Gently pour boiling water over cocoa. Do not stir.
Bake until the top of the cake is cracked, sauce is bubbling and a toothpick inserted into cakey area comes out mostly clean, about 22-24 minutes. Cool on rack for at least 10 minutes. Serve warm with vanilla ice cream.