How to Make It
Heat 2 tablespoons of the oil in a large nonstick skillet over medium-high. Add eggplant and season with ¼ teaspoon of the salt. Cook, stirring often, until softened and golden brown in spots, about 5 minutes. Add mushrooms, shallot, and garlic and cook, stirring often, until mushrooms are golden, 4 to 5 minutes. Transfer to a food processor and let mixture cool at least 5 minutes. Wipe skillet clean.
Add rice, panko, flour, pepper, and remaining ¾ teaspoon salt to food processor. Pulse until mixture comes together, about 15 times. (Mixture will be sticky.) Form into 8 balls (about the size of golf balls).
Heat remaining 1 tablespoon oil in a large nonstick skillet over medium-high. Add meatballs and cook, turning occasionally, until browned on all sides, 8 to 10 minutes. Remove from heat.
Preheat broiler with oven rack 8 inches from heat source. Place toasted rolls on a rimmed baking sheet. Spread ½ cup of the tomato sauce evenly inside rolls. Top with meatballs, remaining ½ cup tomato sauce, and cheese. Broil until cheese is melted and subs are warmed through, 1 to 2 minutes. Top with the torn basil.