Remove and reserve 1 tablespoon sugar. Cook remaining sugar, 1/3 cup butter, and next 3 ingredients in a medium saucepan over medium-low heat, stirring constantly, until butter melts. Cool to 110°.
Stir together yeast, beer, and reserved 1 tablespoon sugar in a 2-cup liquid measuring cup; let stand 5 minutes.
Combine milk mixture and yeast mixture in a large bowl; stir in egg. Gradually stir in 4 cups flour to form a dough. (Dough will be very stiff.)
Turn dough out onto a lightly floured surface, and knead, adding additional flour (up to 1/2 cup) as needed, until smooth and elastic (about 6 to 8 minutes). Place in a bowl coated with cooking spray, turning to coat top of dough. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk.
Coat muffin pans with cooking spray. Punch dough down, and shape dough into 36 (1-inch) balls. Place 3 dough balls in each muffin cup. (Handle the dough as little as possible to prevent over-kneading.) Cover and let rise in a warm place, free from drafts, 1 hour.
Bake at 350° for 15 to 18 minutes or until golden. Place a small pat of butter on top of each roll after baking 5 minutes, or brush with melted butter after baking, if desired.