What is the best way to deal with broken tree branches from icy weather?
It is best to allow the damaged tree to thaw before attempting to work with it as your efforts may produce more damage. Do not attempt to remove ice by striking the branches. Trees are fairly brittle in winter; and with the added rigidity of the ice, you will break more branches than you will save.
The primary factor in your decision for dealing with the tree is safety. Is the tree damaged to the point that removing the limb is a hazardous task? If so, then you should consider hiring a Certified Arborist to help with the situation. Is it a tree that, by nature has weak branches? If the answers are yes, then the best decision may be to remove the tree.