For piperade, heat oven to 450 degrees. Place pepper halves on a foil-lined sheet, cut side down. Roast until skin loosens, about 15 minutes. Remove from heat and let rest until cool enough to handle. Peel and chop finely.
Combine oil, garlic, and onion in medium skillet over low heat until very soft but not browned, about 8 minutes. Add tomatoes, their juices, thyme, parsley, and bay leaf. Simmer over low heat until very soft and very little liquid remains, about 10 minutes, do not brown; add peppers and simmer to soften them. Season to taste with salt, and discard herbs. Reserve tablespoon of mixture and spread remainder in bottom of an 8-inch skillet.
For vegetables, heat oven to 275 degrees. Down center of pan, arrange a strip of 8 alternating slices of vegetables over piperade, overlapping so that only 1/4 inch of each slice is exposed. Around the center strip, overlap vegetables in a close spiral that lets slices mound slightly toward center. Repeat until pan is filled; all vegetables may not be needed.
Mix garlic, oil, and thyme leaves in bowl and season with salt and pepper to taste. Sprinkle over vegetables. Cover pan with foil and crimp edges to seal well. Bake until vegetables are tender when tested with a paring knife, about 2 hours. Uncover and bake for 30 minutes more. (Lightly cover with foil if it starts to brown.) If there is excess liquid in pan, place over medium heat on stove until reduced. (At this point it may be cooled, covered and refrigerated for up to 2 days. Serve cold or reheat in 350-degree oven until warm.).
For vinaigrette, combine reserved piperade, oil, vinegar, herbs, and salt and pepper to taste in a bowl.
To serve, heat broiler and place byaldi underneath until lightly browned (about 5 minutes on low). Slice in quarters and very carefully lift onto plate with offset spatula. Turn spatula 90 degrees, guiding byaldi into fan shape. Drizzle vinaigrette around plate. Serve hot.