What is the name of this eggplant dish that is similar to lasagna?
In Italian we call it parmigiana.  

Wikipedia says:


  Parmigiana or eggplant parmigiana (Italian: melanzane alla parmigiana or parmigiana di melanzane) is a Southern Italian dish made with shallow fried eggplant slices layered with cheese and tomato sauce, then baked. Variations made with breaded meat cutlets, such as veal and chicken parmigiana, have been developed in other countries, usually in areas of Italian immigration.


In Italy, you can find the variation with meatballs, but the original recipe (being from Southern Italy) doesn't include meat.