01 - Preheat the oven to 400°F. Grease a large baking dish with olive oil or butter.
02 - Melt butter in a saucepan over medium heat. Whisk in flour and cook for 1 minute. Gradually add milk, whisking constantly until sauce thickens, about 5 minutes. Season with nutmeg, salt, and pepper. Remove from heat.
03 - Arrange half of the potato slices evenly in the baking dish. Layer half the zucchini, eggplant, bell pepper, and onion on top. Sprinkle with half the chopped parsley, dried thyme, salt, and pepper.
04 - Pour half of the béchamel sauce evenly over the vegetable layers. Repeat layering with remaining potatoes, zucchini, eggplant, bell pepper, onion, herbs, salt, and pepper. Pour remaining béchamel sauce over the final layer.
05 - Combine Gruyère, Parmesan, and breadcrumbs in a small bowl. Sprinkle evenly over the top. Drizzle with olive oil.
06 - Cover the dish loosely with aluminum foil and bake for 25 minutes. Remove foil and bake for an additional 15 minutes until the top is golden and bubbling and the vegetables are tender.
07 - Let the gratin rest for 5 minutes before serving. Garnish with fresh parsley.