Cut each eggplant in half lengthwise, leaving the stem intact. Scoop out soft insides, leaving 1/2 inch shell all around.
Put rice in cooker. Add broth and water and 1/2 teaspoon salt. Stir. Cook 35-40 minutes, till done.
Heat the oil in a medium frying pan.
Saute the shallot till softened.
Add onion and saute till just beginning to turn golden brown
Add mushrooms and saute till liquidy.
Add salt, pepper, and basil and cook till liquid evaporates.
Stir in scooped out eggplant. Cook for 5 minutes over low heat.
Combine with cooked rice, mixing well.
Stuff reserved eggplant shells.
Place on baking sheet. Sprinkle each eggplant half with a teaspoon of bread crumbs and bake at 350F for 15-20 minutes.