Spread a small amount of tomato sauce on the bottom of a 13 × 9 inch baking dish. Arrange half the lasagne noodles over the sauce. Cover with the peppers, onion, mushrooms, and eggplant, in layers. Mix the béchamel and Parmesan together, and spoon half on top of the vegetables. Crumble the chèvre over the béchamel, and sprinkle with half the basil. Top with more tomato sauce, and then another layer of lasagne noodles. Spread the remaining béchamel over the lasagne, and sprinkle with the remaining basil. Top with the remaining tomato sauce. Sprinkle with the mozzarella.