Preheat oven to 350 degrees F.
Place the potatoes in a medium pot and cover with water by 1-inch. Bring to a boil and cook until just tender, about 15 minutes. Drain and when cool enough to handle, remove the skins. Slice horizontally into 1/2-inch thick slices.
Peel stems of cardoons with a vegetable peeler. Discard leafy tops. Blanch peeled cardoons in a pot of boiling water with milk, until tender, about 3 minutes. Remove and cool. Cut into 1/2-inch pieces.
In a small bowl, combine thyme, garlic, red pepper flakes, salt, and pepper.
In a large oval casserole dish, alternately layer at an angle the potatoes, mozzarella, peppers, tomatoes, onions, and zucchini, seasoning between each layer with garlic, thyme, red pepper mixture, salt, and pepper. Drizzle with the olive oil and bake until the vegetables are tender and beginning to turn golden brown, 35 to 40 minutes. Sprinkle the cheese over the top and return to the oven until the cheese is melted, 5 to 8 minutes.
Remove from the oven and serve hot.
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