I would not pull that broken tooth. You are inviting a problem. I would do a gold post and crown. First endo is done on the tooth to remove the pulp chamber and the neurovascular bundle in the root that feeds the tooth from the primary above or below it with blood. Then the dentist will drill out a small receptor site down into the remaining portion of the tooth and root and make a direct pattern of it using a self-curing resin. This will be sent to a laboratory who will add some wax to the supragingival part of it making it a shape kind of like a crown prep on a natural tooth. Then you will revisit the dentist and he will cement this gold post into the root, and a crown can be made over the whole thing. Even though the tooth will be dead, it will be come ankelosed into the bone, with the fibers that normally hold it into place desiccating, and it will be safely and permanently there as long as you keep it clean and do not let periodontal disease creep in. All this circumvents an extraction that could lead to a non-healing socket due to lack of blood supply and micro vascularization