Whisk together 2 eggs. Grease a large bowl with vegetable oil and set aside. Combine flour, sugar, salt, yeast and softened butter in the bowl of an electric mixer fitted with either the paddle attachment or dough hook. Turn speed to low. Gradually pour in hot milk and water while mixing. Add beaten eggs. Beat on low for a total of 12 minutes.
Remove from mixer and knead by hand for 1 minute or until smooth and elastic. Shape into a ball and place into greased bowl. Turn dough upside down in bowl so that the top is now greased. Cover with a damp kitchen towel or plastic wrap. Place in a warm, draft-free environment and allow dough to rise until doubled in size, approximately 1 1/2 hours. Punch dough down gently and turn onto a lightly floured surface. Form into a smooth ball. Let rest for 15 minutes covered with inverted bowl used for rising. Proceed with specific shaping instructions below.
Once the rolls have been shaped, cover with a damp towel, wax paper or aluminum foil (the covering should not touch dough -- it may have to be "etented"e over the rolls) and place in a warm, draft-free environment until rolls double in size, about 40 minutes. Heat oven to 400°. Bake rolls for 18 to 20 minutes. Turn pan halfway through baking. Tops should be golden brown
Parker House Rolls: Melt 8 tablespoons (one stick) butter in a saucepan. Set aside. On a lightly floured surface, roll out dough to a 1/2-inch thickness. Cut circles using a 2 3/4-inch floured biscuit cutter. Working with one piece of dough at a time, shake off excess dough and dredge in butter. Fold each piece in half and place in pan, flat side down and in rows, slightly overlapping last piece. Yields 36 rolls
Crescent rolls: Roll 1/4 of the dough into a 12-inch circle and cut into eight wedges, like a pie. Roll up each piece from the perimeter to the tip. Curve ends to make crescents. Repeat with remaining dough. When placing rolls on baking sheet, place tip of roll underneath and brush with melted butter. Yields 32 rolls.
Cloverleaf rolls: Gently roll dough into 1-inch balls. Grease a muffin tin and drop three balls into each tin. Brush with melted butter. Yields 36 rolls
Fan rolls: Roll the dough into a 3/8-inch sheet and then cut into 1 3/4-inch strips. Make piles of 5-strips each and then cut into squares. Place squares on edge in greased muffin tins. Brush with melted butter. Yields 30 rolls.