Preheat the oven to 350 degrees F.
Cut out the stems from peppers and discard. Remove the seeds and clean out the white ribs. Season them inside with salt and pepper. Rub the outsides of the peppers with vegetable oil and place them, stem side down, onto a baking sheet. Bake them for 30 minutes, or until the skins are wrinkled and slightly charred. Remove the peppers from the oven and put them into a large bowl. Cover the bowl with plastic wrap and let them steam until they are cool enough to handle, about 10 minutes. Peel the skin from the peppers, making sure to keep them whole, and set aside.
While the peppers are baking and steaming, make the pesto: Bring a large pot of water to a boil. Have a bowl of ice water ready. When the water is boiling, add the basil leaves. After 5 seconds (yes, 5 seconds) remove the basil and immediately put it into the ice water. (This will keep the pesto green.) When it is cold, drain the basil and squeeze out any excess water. Put the basil into a food processor or blender and add the pine nuts, cheese, and garlic. Turn on the processor and slowly add the olive oil until the mixture is smooth. Refrigerate until ready to use.
In a large skillet over high heat, melt 2 tablespoons of the butter. Cook the eggplant until beginning to soften, about 5 minutes. Add the onions and cook until softened and lightly browned, about 10 minutes more. Transfer them to a cutting board, let them cool a bit, and chop them finely with a knife. Add 1 tablespoon butter to the same pan over medium-high heat. When it is melted, add the Swiss chard, cover the pan, and let the chard wilt for about 1 minute. Transfer the chard to a bowl. Add the vegetables, ricotta, 1 tablespoon of the pesto, and season with salt and pepper. Mix well, taste, and adjust the seasoning. Stuff this mixture into the peppers, being careful to keep them whole. Place them on their sides into a lightly buttered baking dish. Sprinkle each pepper evenly with bread crumbs and bake until the tops are golden brown, about 10 minutes. Serve immediately garnished with extra pesto.