1) Slice courgettes and yellow squash lengthwise into 0.25 to 0.5cm thick strips. Lay the strips out on baking sheet pan and salt generously. Set aside.
2) Roast the peppers over a gas flame or griddle on a baking sheet until completely charred, about 8 minutes. Put the peppers in a bowl, cover, and set aside until just cool enough to handle, about 5 minutes. Rub the skins off the peppers with your fingers to remove most of the charred skin. Core and remove the seeds, then slice the peppers lengthwise into 3cm strips.
3) Preheat the oven to 190C/Gas 5. Mix the goat cheese, olives, thyme, and red pepper flakes together in a bowl and season, to taste, with salt and pepper.
4) Thoroughly wipe excess salt and moisture from courgettes and squash with paper towel.
5) Spread 100g of the tomato sauce into the bottom of a 20 by 33cm baking dish. Begin layering, remembering to season with pepper between each layer. Start with the courgettes and then summer squash and all the peppers. Dollop spoonfuls of the cheese over the peppers and then spread it out to cover.
6) Spoon half of the remaining sauce over the vegetables. Repeat with the remaining courgettes, squash, goat cheese, and sauce. Sprinkle top layer with Parmesan and bake, covered, for 30 minutes, then uncover. Continue baking until the vegetables are tender and the sauce thickens, about 30 minutes more. Let lasagna stand for 10 minutes before slicing.
For tomato-basil sauce
1) Put the tomatoes, olive oil, garlic, vinegar, and salt in a sauce pan. Bring to a boil over high heat, lower heat to a brisk simmer, and cook until thickened, about 30 minutes.
2) While sauce cooks tear basil leaves into small pieces. Remove the sauce from the heat and stir in the basil. Transfer half of the sauce to a food processor and puree. Return to the saucepan and season, to taste, with salt and pepper.