Preheat the oven to 350 degrees F. Cut the chayote in half lengthwise. Boil in cold salted water, and then reduce heat. Let simmer covered for 12-15 minutes (until tender) and drain. When the chayote is cool enough to handle, de-seed it. Scoop out the pulp, then drain and chop it in a separate bowl. (If the pulp is too wet, squeeze between paper towels to remove extra liquid)
In a large skillet, cook mushrooms, sweet pepper, onion, and garlic with butter. Remove from heat when they are tender, do not brown them. Stir chayote pulp into the mixture and add toasted bread crumbs, 1/3 cup of parmesan cheese, egg, parsley (or cilantro), and salt and pepper.
Put the bouillon cubes in the water so that they dissolve; when dissolved, add to the stuffing. Spoon stuffing into the chayote. Put the shells in a baking dish, cover and bake for about 25 minutes. When finished, sprinkle the remainder of the parmesan cheese on the top and bake for about 3 minutes until the cheese melts.