This recipe is not an exact clone, since she always kept it a secret. But a little experimentation came up with something very, very close.
Mirlitons are also known as "chayote squash". If you can't find them, eggplant would also be delicious.
First you cut up the mirlitons and cook them separately. I boiled them, but you can also smother them for hours in onions and a small amount of cooking oil. When they're done, mash them up, drain, and set aside.
Melt some butter in a cast iron skillet and cook some chopped onions, minced garlic, and shrimp. Add garlic powder, onion powder*, oregano, thyme and black pepper.
*I call garlic powder and onion powder "the dynamic duo." That goes in almost everything I cook.
When that's cooked, drain the excess liquid, add in the mashed mirlitons, and mix.
Add italian bread crumbs and parmesan cheese, mix all the way through, then add extra of both on top. Add a few small pats of butter. Bake at 350.