Position an oven rack 4 inches from the broiler and preheat the broiler to high.
Whisk together 1/4 cup of the oil, the vinegar and mustard in a 9- by 13-inch baking dish. Season with salt and pepper. Dip each mushroom cap in the marinade, turning to coat on all sides. Let the mushrooms sit in the marinade for 15 to 20 minutes.
Heat the remaining 2 tablespoons oil in a large nonstick skillet over medium-high heat. Add the shallots and saute until just beginning to soften, about 2 minutes. Add the garlic and red pepper flakes and cook for 1 minute. Add the squash, zucchini and tomatoes, sprinkle with salt and pepper and saute until beginning to soften, 4 minutes.
Mix together the panko, Parmesan and parsley in a small bowl. Season with salt and pepper and set aside.
Heat a grill pan over medium-high heat. Grill the mushrooms, cap-side down, until slightly charred, about 3 minutes per side. Wipe out the baking dish with a paper towel and return the mushrooms to the baking dish, cap-side down. Divide the sauteed vegetables among the mushroom caps. Cover the vegetables with the panko mixture, drizzle with a bit of olive oil and place under the broiler until browned, about 2 minutes. Serve hot or at room temperature.
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