Preheat oven to 325 degrees F (165 degrees C). Grease a 9x13-inch baking pan.
Combine egg yolks and sugar in a bowl; beat with an electric mixer until pale yellow and creamy. Drizzle butter and water into the bowl, beating until incorporated, 1 to 2 minutes. Beat in flour. Pour in milk; beat on low speed until batter is blended.
Wash beaters and dry well. Beat egg whites in a glass, metal, or ceramic bowl until stiff peaks form. Fold egg whites into the batter, a third at a time, until no white streaks remain. Pour batter into the prepared baking pan.
Bake in the preheated oven until top is lightly browned, 1 hour to 1 hour 10 minutes. Cool completely before cutting into squares, about 30 minutes.