Stir milk, 4 1/2 tablespoons shortening, sugar, and salt together in a bowl. Cool until mixture is lukewarm.
Pour water into a bowl and sprinkle yeast over the top. Let stand until yeast softens and begins to form a creamy foam, about 5 minutes; stir. Stir milk mixture into yeast mixture until smooth.
Beat 2 1/4 cups flour into yeast mixture until smooth. Gradually stir in remaining flour until dough comes together.
Turn dough out onto a lightly floured work surface and knead until dough is smooth and elastic, about 8 minutes. Place dough in an oiled bowl, brush with melted shortening, cover the bowl with a clean towel, and set in a warm place to rise until doubled in volume, about 2 hours.
Punch dough down and form into 24 rolls. Place rolls on a lightly greased baking dish, cover with a towel, and let rise until doubled in volume, about 40 minutes.
Preheat oven to 350 degrees F (175 degrees C).
Bake in the preheated oven until golden brown, 12 to 15 minutes.
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