Place pumpkin, water, milk, egg, oil, flour, sugar, salt, pumpkin pie spice, and yeast into a bread machine pan in the order suggested by the manufacturer. Select Dough setting and press Start.
Turn dough out onto a lightly floured work surface. Shape into dinner rolls and place on lightly greased baking sheets. Cover with a towel and let rise until almost doubled in size, about 45 minutes.
Preheat the oven to 375 degrees F (190 degrees C).
Bake in the preheated oven until rolls are golden brown, about 15 minutes. Remove from the oven and immediately remove rolls from pans and cool on a wire rack to prevent crusts from becoming soggy.