Slice zucchini and yellow squash lengthwise into 1/8 to 1/4-inch thick strips. Lay the strips out on baking sheet pan and salt generously. Set aside.
Roast the peppers over a gas flame or broil on a sheet pan 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 1 1/2-inch strips.
Preheat the oven to 375 degrees F. Mix the goat cheese, olives, thyme, and red pepper flakes together in a bowl and season, to taste, with salt and pepper.
Thoroughly wipe excess salt and moisture from zucchini and squash with paper towel.
Spread 1 cup of the tomato sauce into the bottom of an 8- by 11-inch baking dish. Begin layering, remembering to season with pepper between each layer. Start with the zucchini and then summer squash and all the peppers. Dollop spoonfuls of the cheese over the peppers and then spread it out to cover. Spoon half of the remaining sauce over the vegetables. Repeat with the remaining zucchini, 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.
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. 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.
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