Preheat the grill.
Before preparing the vegetables, roast the garlic. Rub 1 teaspoon of the oil and a pinch of the salt and pepper onto the cut side of the garlic. Wrap in foil and place on the grill. Roast until soft and starting to caramelize, turning occasionally, about 25 minutes. Remove from the grill and let sit until cool enough to handle. Squeeze the pulp from the garlic skins and set aside.
Rub 2 teaspoons of the oil and pinch of the salt and pinch of the pepper on the tomatoes. Place on the grill away from the direct heat (or on the upper rack) and grill until tender and start to char, about 6 minutes, depending upon the heat. (The longer the tomatoes stay on the grill, the smokier flavor they will obtain.) Remove from the heat and when cool, remove the skin and roughly chop.
In a large bowl, combine the eggplant, zucchini, squash, onions, and bell peppers and toss with 2 tablespoons of the oil, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper. Grill, turning, until marked by the grill and tender, 3 to 5 minutes per side. Remove with tongs and let cool. Dice into 1/2-inch pieces.
In the bottom of a large bowl, mash the roasted garlic and 1 teaspoon of olive oil to make a paste. Drizzle in the remaining 1 1/2 tablespoons of oil, 1 teaspoon salt and 1 teaspoon pepper, and the vinegar and whisk to combine. Add the chopped vegetables, tomatoes, and the herbs and toss to coat. Serve at room temperature, garnished with Parmesan. (The ratatouille can be made ahead and refrigerated overnight to allow the flavors to further develop.)
Serve atop grilled wild Alaskan salmon.