Preheat oven to 350°F.
Heat about 2 tablespoons of canola oil in a large skillet over medium heat. When hot, reduce heat to medium-low, add onions and garlic, sprinkle with salt and add 1 1/2 teaspoons of thyme. Stirring occasionally, cook until tender, about 20 minutes.
Slice squash, zucchini, and eggplant into 1/4 inch thick rounds and place in a large bowl. Add 1/4 cup olive oil and toss to coat. Sprinkle with salt and pepper and toss again.
Slice Roma tomatoes 1/4 inch thick and place rounds overlapping on a plate. Drizzle with olive oil and sprinkle with salt and pepper.
In a small bowl, mix together grated cheese, bread crumbs, and 1/2 teaspoon dried thyme.
Spread onion mixture on the bottom of a 9x13 inch (oval) gratin pan or an 8x8 inch baking pan.
Working on the diagonal, place a row of overlapping zucchini in the corner of the pan. Sprinkle with cheese mixture, followed by a rows of yellow squash, tomatoes, and eggplant, each overlapping the previous row, and sprinkle with cheese after finishing each row (see picture below).
Continue placing rows of vegetables in the pan until the onions are completely covered. Sprinkle with salt and pepper, followed by the remaining cheese mixture.
Bake for 1 to 1 1/2 hours or until the vegetables are very soft and easily pierced with a sharp knife. Let stand for 10 minutes.
If desired, place gratin under broiler to brown.