Arrange the pastry in a pie plate, pressing firmly on the bottom and sides. Fold any overhanging pieces of pie dough under itself and crimp the edges with your fingers (or the tines of a fork) to make a pretty edge. Place a piece of parchment paper on top of the crust and fill with pie weights or dried beans. Bake for 8-9 minutes until pastry is set. Remove from oven and let cool.
Reduce the heat to 350°.
Place the chopped eggplant on a baking sheet. Sprinkle with half the rosemary and thyme and drizzle with 1 tablespoon of the olive oil. Use your hands to toss the herbs and oil with the eggplant until it's coated. Arrange the eggplant on a single layer and bake in the hot oven for 15 minutes. While the eggplant is cooking, place the bell pepper in the oven to roast for 30 minutes or until skin is blackened and flesh is tender.
While the eggplant roasts, toss the chopped onions with the remaining olive oil and herbs. Add the onions to the eggplant after the first 15 minutes and toss to combine. Spread into a single layer and continue to cook for an additional 15 minutes, until eggplant has shrunk and appears a little dry and the onions are tender. Remove from oven and let cool.
Remove the bell pepper from onion and place in a small bowl. Cover with plastic wrap and steam for about 10 minutes.
Slice the tomato crosswise and remove the seeds. Lay the slices on a cutting board and lightly sprinkle with salt. Set aside.
When pepper is cool enough to handle, peel the skins and discard them. Remove the seeds and slice the pepper into thin strips. Reserve 5 strips for garnish, and chop the remainder.
In a medium bowl whisk together the eggs and milk. Stir in the 3/4 of shredded cheese and basil. Add the chopped pepper and roasted eggplant and onion mixture. Stir to combine. Pour the eggplant mixture into the prepared pastry.
Pat the tomato slices dry with a paper towel and lay the sliced tomatoes on top of the quiche, pressing them slightly into the egg mixture. Arrange the slices of pepper in between the tomato slices. Top with the remainder of cheese and bake for 20 minutes. Turn the dish 180° in the oven and bake for an additional 20 minutes. Turn the dish one more time and bake for 10 more minutes. Let the quiche rest for about 20 minutes before serving.