Add the water, milk, ⅓ cup melted butter, sugar, egg, flour, salt, and yeast to the bread machine pan in the order listed. Make a small well in the flour for the yeast, keeping it from touching the wet ingredients.
Select the dough cycle and press start.
Once the dough cycle is complete, lightly grease a 9x13-inch baking pan. Lightly flour a cutting board or work surface and remove the dough from the bread machine.
Divide the dough into 20 equal pieces for dinner rolls or 12 larger pieces for hamburger buns.
Roll each piece into a smooth dough ball and place seam-side down in the prepared baking pan.
Brush the tops with half of the 4 tablespoons melted butter, reserving the rest. Cover and allow the rolls to rise in a warm place for about 45 minutes.
Preheat the oven to 375°F. Bake for 16-18 minutes, or until lightly golden brown. Remove from the oven and immediately brush with the remaining melted butter. Serve warm and enjoy!
Notes
Bread flour can be substituted for all-purpose flour for a slightly chewier texture.
Bread machine brands vary, so check your manufacturer's instructions for the recommended ingredient order and dough cycle.
The dough should form a soft, smooth ball during the dough cycle. If it is overly sticky, add a small amount of flour; if it seems too dry, add a small amount of water.
Rise time can vary depending on the temperature of your kitchen. The shaped rolls may need a little longer than 45 minutes in a cooler room.
For the softest rolls, avoid overbaking and brush with melted butter as soon as they come out of the oven.