Make the cinnamon rolls: In a saucepan, heat milk, water, and 2 tablespoons butter until very warm. Pour mixture into a bread machine.

Add flour, 1/4 cup sugar, egg, salt, and yeast to the bread machine in the order suggested by the manufacturer. Run Dough cycle.

Once Dough cycle is complete, remove dough from the bread machine and punch down. Roll into a large rectangle on a floured surface. Smear with softened butter.

Grease a 9x13-inch baking pan. Combine 1/3 cup sugar and cinnamon together in a small bowl.

Sprinkle sugar mixture over dough, then generously sprinkle with raisins and walnuts. Starting at the long edge, roll dough into a log, then cut into 12 equal pieces. Place rolls, cut-side down, into the prepared baking pan. Cover and let rise in a warm place until almost doubled in volume, about 30 minutes.

Preheat the oven to 375 degrees F (190 degrees C).

Bake rolls in the preheated oven until golden, 20 to 25 minutes. Remove and allow to cool for 10 minutes.

While the rolls are cooling, make the frosting: Mix confectioners' sugar, melted butter, milk, and vanilla together in a bowl until thick.

Spread frosting over warm rolls.