Heat olive oil in a large pot or Dutch oven over medium heat. Cook and stir onion until it begins to soften and turn translucent, about 5 minutes. Add eggplant and cook until they have absorbed some of the oil, about 1 minute. Stir zucchini into eggplant mixture and cook until softened, 1 to 2 minutes. Mix in mushrooms and salt; cook and stir until mushrooms are softened, 3 to 4 minutes.
Pour Marsala wine into eggplant mixture and bring to a boil, scraping up any browned bits from the bottom of the pan. Cook until wine has reduced slightly, 2 to 3 minutes.
Spread each ciabatta roll with 1 tablespoon garlic basil spread, 1 tablespoon garlic roasted tomato sauce, and 1 tablespoon artichoke aioli. Top with 1/4 pound mozzarella cheese. Evenly distribute the eggplant mixture among the 4 sandwiches.
Place sandwiches on a baking sheet and bake in the preheated oven until the cheese is melted and the bread is toasted, 3 to 4 minutes.