Prepare the Lemon Brownies: Preheat oven to 350°F. Line an 8-inch-square baking pan with parchment, and lightly grease (with cooking spray). Set aside.

Beat sugar and butter in a large bowl with an electric mixer on medium-high speed until light and fluffy, 1 1/2 to 2 minutes. Add eggs, 1 at a time, beating on medium speed until fully incorporated after each addition. Beat in lemon zest and juice. Beat in white chocolate. Whisk together flour and baking powder in a small bowl. Fold flour mixture into butter mixture until fully incorporated.

Pour batter into prepared baking pan, and smooth to edges. Bake in preheated oven until a toothpick inserted into center comes out with a few moist crumbs, 22 to 25 minutes. Let cool completely in pan on a wire rack, about 1 hour.

While brownies cool, prepare the Lemon Frosting: Beat cream cheese in a large bowl with an electric mixer on medium speed until fluffy, about 1 minute. Beat in powdered sugar on low speed until fully incorporated. Add lemon juice and 1 teaspoon milk; beat on high speed until smooth, about 1 minute, adding remaining 1 teaspoon milk, if needed. Spread over cooled brownies. Cut brownies into squares, and serve.