Heat 1/2 tablespoon oil in each of 2 large nonstick skillets over moderate heat. Divide onion between skillets and cook, stirring occasionally, until golden brown, caramelized and very soft, about 20 minutes. Combine both onion batches in one of the skillets. Add 3 tablespoons vinegar and the thyme sprigs. Cook, stirring, until vinegar has evaporated, about 1 minute more. Discard sprigs; salt and pepper to taste. Cool to room temperature, stirring occasionally, about 20 minutes.

Meanwhile, preheat oven to 500 degree F and arrange rack in lower third. Lightly oil a large cookie sheet. Using a floured rolling pin, roll out dough on a lightly floured surface into a 17-by-13-1/2-inch rectangle, then roll in edges to form a 16-by-12-1/2-inch crust. Transfer to prepared sheet and evenly spread onion mixture over dough, avoiding edges. Decoratively arrange olives and pepper strips over top.

Bake until crust is golden brown on bottom, about 18 minutes. Cut pizza into 10 squares.

Make salad: While pizza is baking, whisk together remaining 1 tablespoon oil and 2 tablespoons vinegar until thickened and emulsified. Toss with cheese and salad mix; salt and pepper to taste. Serve with pizza. Makes 10 servings.