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1. Cut 1 squash, 1 zucchini, and 1 carrot into 1/4-inch-thick slices. Place in a Dutch oven; add onion, garlic, 1/2 cup vegetable broth, 3 Tbsp. olive oil, and 1/2 tsp. salt. Cover and cook over medium-low heat, stirring occasionally, 20 to 30 minutes or until vegetables are very tender. Stir in 1/4 cup basil; cool 10 minutes.
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2. Meanwhile, saute prosciutto in a lightly greased large nonstick skillet over medium heat 6 to 8 minutes or until browned and crisp; remove from skillet. Wipe skillet clean.
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3. Process cooked squash mixture and remaining 1/4 cup broth in a blender or food processor until smooth. Wipe Dutch oven clean.
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4. Cook pasta in Dutch oven according to package directions; drain, reserving 1 cup hot pasta water. Return hot cooked pasta to Dutch oven.
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5. Cut remaining squash, zucchini, and carrot lengthwise into very thin, ribbon-like strips using a mandoline or Y-shaped vegetable peeler. Stack ribbons, and cut in half lengthwise.
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6. Melt butter with 1 Tbsp. olive oil in skillet over medium heat; add vegetable ribbons, green onions, and remaining 1/2 tsp. salt, and saute 5 minutes or just until tender. Transfer to a plate, and cover.
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7. Cook squash mixture, mascarpone cheese, and 1/4 cup Parmesan cheese 3 to 4 minutes or just until sauce is hot and cheese is melted. Pour sauce over pasta; toss to coat, adding desired amount of reserved hot pasta water to thin sauce, if necessary. Top with vegetable ribbons, prosciutto, and remaining 1/2 cup Parmesan cheese and 1/4 cup basil. Drizzle with remaining 1 Tbsp. olive oil.
