Heat 2 tablespoons olive oil in a skillet over high heat. Cook and stir onion until browned, 3 to 5 minutes. Add bell peppers and reduce heat to low. Cook and stir for 5 minutes. Pour in vegetable broth. Cook and stir for 10 minutes.
Meanwhile, heat remaining 2 tablespoons olive oil in a large skillet over medium heat. Quickly brown eggplant slices - they should be tender enough to roll easily, but not so tender they break apart.
Transfer cooked red bell pepper mixture to the bowl of a food processor; pulse until smooth. Season with salt and pepper and set aside.
Preheat the oven to 400 degrees F (200 degrees C).
Combine mozzarella cheese, tomatoes, and basil in a bowl. Season with salt and pepper and mix well.
Lay eggplant slices on a sheet of parchment paper. Spoon a little of the cheese mixture onto 1 end of an eggplant slice and roll up. Place eggplant roll in a casserole dish. Repeat with remaining eggplant and cheese mixture.
Bake in the preheated oven until cheese is melted, about 10 minutes. Remove from oven. Serve with warm bell pepper sauce.