In a large bowl, stir together 2 cups flour, sugar, shortening, salt and yeast. Add water, milk and egg. Beat with electric mixer on low speed 1 minute, scraping bowl frequently. Add the remaining flour, 1/4 cup at a time, until the dough comes together.
Turn the dough out onto a lightly floured surface and knead until smooth and elastic, about 8 minutes. Place in greased bowl and turn greased side up. Cover and let rise in warm place until doubled, about 1 hour. Dough is ready if indention remains when touched.
Turn dough out onto a lightly floured surface and divide in half; cut each half into 24 pieces. Form the pieces into rounds. Place the rolls close together in two lightly greased 9 inch round cake pans. Brush with melted butter. Cover and let rise in warm place about 30 minutes or until doubled. Cover the rolls with a damp cloth and let rise until doubled in volume, about 40 minutes. Meanwhile, preheat oven to 400 degrees F (200 degrees C).
Bake in preheated oven for 12 to 18 minutes, until golden brown.