Combine milk, shortening, sugar, and salt in a saucepan and cook, stirring constantly, over medium heat until sugar dissolves. Remove from heat; set aside and allow to cool.
Mix yeast with 1/4 cup lukewarm water in a large bowl; set aside until yeast dissolves and little bubbles begin to appear on the surface, about 10 minutes.
Pour cooled milk mixture into yeast. Stir in eggs and gradually add flour, stirring with a wooden spoon until dough is too stiff to stir, then use your hands to knead in the flour (dough will be sticky, so grease your hands with a little butter). Brush a small amount of butter on the inside of a large bowl and on one side of a sheet of waxed paper. Place dough in bowl, cover with buttered waxed paper, and lay a clean damp dish towel on top. Set aside to rise until doubled, at least three hours.
Turn dough out onto a lightly floured surface. Knead until elastic, then roll out to 1/2 -inch thickness. Cut dough with a 3-inch biscuit cutter, dip each round into melted butter, and fold in half. Line up, round edges up, sides touching, in a baking pan. Cover with buttered waxed paper and a towel, and set aside to rise, at least 2 1/2 hours.
Preheat oven to 350 degrees. Bake until golden, about 15 minutes. Rolls are best served warm.
Recipe Courtesy of Saveur Magazine Cookbook: Saveur Cooks Authentic American