My new neighbor has an exclusive ingress/egress easement across my property.

You can hire an attorney to write to your neighbor explaining the law of easements, including both parties' rights and responsibilities. Your attorney can demand the removal of the encroaching unlawful portion of the concrete slab as well as maintenance of the easement in lieu of forfeiture of the associated rights of ingress/egress. Your neighbor's rights are limited to ingress and egress. That is, the scope of the easement is limited to getting in and out and running utility lines. There is no right to construct buildings or other structures in this kind of easement.

Your property is what is called the servient estate. This means your prpoerty is subject to his use under the terms of the easement. A very common problem is when the people who created the easement have sold or dies, and the easement binds the prpoerty but there is no context. It sounds like the use of the easement is substnaitally greater than what was intnended. The rememdy is a cease and desist letter, and if he won't go for that, filing suit to quiet title and for injunctive releif. You should talk to a lawyer. These types of issues usually go the wrong way, and a misused easement can become an adverse possession issue if it goies on long enough.