Preheat oven to 350°F with rack in middle position. Line an 18- x 13-inch rimmed baking sheet with parchment paper or aluminum foil; grease with oil. Stir together panko, cheese, thyme, 1 teaspoon salt, and ½ teaspoon pepper in a small bowl; set aside.

Trim and remove stem end and outer layer of onion, and discard. Using a sharp knife or a mandoline, slice onion into wafer-thin slices (about 1/16-inch thick). Set aside.

Arrange Yukon gold potato slices in lengthwise rows to create a roughly 13- x 9-inch rectangle on prepared baking sheet, overlapping slices slightly. Sprinkle evenly with ¼ teaspoon salt, 1 pinch of pepper, and ¼ cup loosely packed panko mixture. Repeat layering process with onion, sweet potato, zucchini, and yellow squash slices, overlapping slightly and sprinkling with ¼ teaspoon salt, 1 pinch of pepper, and ¼ cup loosely packed panko mixture between layers. Top with tomato slices, overlapping slightly. Gently press down to even out layers. Sprinkle evenly with remaining ¼ teaspoon salt. Top with a piece of parchment paper; cover sheet pan tightly with foil. Bake in preheated oven until a paring knife inserted in gratin meets only slight resistance, about 1 hour.

Remove from oven; remove and discard foil. Sprinkle gratin evenly with remaining ½ cup panko mixture. Rotate pan from front to back, and return to oven on middle rack. Bake, uncovered, at 350°F until a knife inserted in gratin goes in and comes out easily, about 20 minutes. Increase heat to broil. (Do not remove gratin from oven.) Broil until panko is lightly browned, 2 to 3 minutes. Remove from oven; let cool 20 minutes.