In a large, straight-sided sauté pan over low heat, heat 1/4 cup of the olive oil. Add the onion and eggplant, raise the heat to medium, and cook, stirring to distribute the oil, until the eggplant has soaked up the oil and is starting to change color, about 5 minutes.
Add the remaining 1/4 cup oil along with the zucchini and cook, stirring occasionally, until the zucchini starts to change color, about 10 minutes. Season with salt and pepper to taste.
Add the broth, garlic, and herb bouquet. Simmer, uncovered, stirring occasionally and scraping up any browned bits until almost no more liquid remains, about 10 minutes; if needed, turn up the heat to reduce the liquid.
Take the pan off the heat, remove the herb bouquet, and adjust the seasonings. Let the compote rest in the pan for 15 minutes or so before serving.