Place water, milk, 1/3 cup butter, egg, flour, sugar, salt, and yeast into the pan of a bread machine in the order listed, or follow the order recommended by your manufacturer if different. Select Dough/Knead and First Rise Cycle; press start.

When the cycle finishes, turn dough out onto a lightly floured surface. Divide dough in half. Roll each half into a 12-inch circle.

Spread 1/4 cup softened butter over entire round. Cut each circle into 8 wedges. Roll wedges starting at the wide end; roll gently but tightly. Place point side down on ungreased cookie sheet. Cover with a clean kitchen towel and put in a warm place to let rise for 1 hour.

Meanwhile, preheat the oven to 400 degrees F (200 degrees C).

Bake rolls in the preheated oven until golden, 10 to 15 minutes.