In a small saucepan, combine the milk, water, and the butter; heat until very warm (about 120 degrees) and butter melts. Add the egg.
In a large bowl of electric mixer, combine 2 cups of the flour with the sugar, yeast, and salt. Stir well. Add milk-egg mixture to the flour mixture. Beat at medium speed until batter is smooth, about 2 minutes. Add enough remaining flour to make a soft dough. Turn out onto floured surface and knead until smooth, and elastic, about 5 to 8 minutes. Place dough in a greased bowl, turning once to grease the top of the dough. Cover with a damp towel and let rise in a warm place for 1 hour until it doubles in size.
Meanwhile, combine the brown sugar and cinnamon in a small bowl and melt the butter. Set aside.
Punch dough down. Cover again and let rest 10 minutes more. Divide dough in half and set one-half aside. Roll the other half out onto a floured surface to form an 18x12-inch rectangle; brush with half of the melted butter, leaving a 1/2-inch border. Sprinkle with half of the brown sugar and cinnamon mixture. Beginning at the long side, roll up jellyroll fashion; moisten edges with water and press together to seal. Cut jellyroll into 1 1/2-inch slices; place cut sides of slices (rolls) down in a greased 9x13-inch baking pan (12 rolls to a pan). Cover with a damp towel and let rise in a warm place for 45 minutes to an hour or until it doubles in size. Repeat with remaining half of the dough.
When rolls have risen, bake them in a preheated 375 degrees oven for 20 minutes or until golden brown. While rolls are baking, prepare the glaze.
In a small saucepan, heat together the butter and milk until the butter is melted. (You can use a glass measuring cup in the microwave to do this.) Add the vanilla and enough of the powdered sugar for desired spreading consistency. Apply glaze when rolls have cooled slightly.