Preheat the oven to 425 degrees.
Cut the fennel bulbs into 6 wedges each, cutting through the core to keep the wedges intact. Place on a sheet pan. Cut the potatoes in half lengthwise and place them on the pan with the fennel. Drizzle the olive oil on the vegetables, then sprinkle with 2 teaspoons salt and 1 teaspoon pepper. Toss with your hands.
Roast the vegetables for 25 to 30 minutes, until the potatoes are tender, tossing once while cooking. Toss the string beans and asparagus with the roasted vegetables and roast for another 10 to 15 minutes, until the green vegetables are tender. Sprinkle on the Parmesan cheese and roast for another minute or two until the cheese melts.
Sprinkle with salt and pepper and serve hot.
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