Preheat the oven to 400 degrees F (200 degrees C). Brush a 9-inch square baking dish with 2 tablespoons of the melted butter.

To make the dough: Whisk flour, 1/4 cup white sugar, baking powder, and salt together in a large bowl. Work 3 tablespoons softened butter into flour mixture using your hands.

Beat milk and egg together in another bowl; pour into flour-butter mixture and stir with a rubber spatula until a soft dough forms. Turn dough out onto a well-floured work surface and roll dough into a 1/4-inch thick rectangle. Brush surface of dough with remaining 2 tablespoons melted butter.

To make the filling: Whisk 1/2 cup white sugar, brown sugar, and cinnamon together in a small bowl. Sprinkle 1/2 of the cinnamon sugar mixture in the bottom of the prepared baking dish. Sprinkle remaining cinnamon sugar over butter-brushed dough.

Roll dough around filling to form a log; cut log into 15 rolls and place rolls in the prepared baking dish. Bake in the preheated oven until rolls are set, 20 to 25 minutes.

To make the frosting: Beat confectioners' sugar, cream cheese, 1/4 cup softened butter, and vanilla extract together in a bowl until frosting is smooth. Top hot cinnamon rolls with cream cheese frosting.

Cover leftover frosted or unfrosted rolls tightly and store at room temperature for up to 2 days or in the refrigerator for up to 5 days.