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Preheat oven to 400 degrees F.
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Cook pasta according to package directions; reserve 4 Tbsp. pasta water for use later.
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On baking sheet, spread out tomatoes and drizzle with olive oil; sprinkle on salt.
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On wire baking rack, lay prosciutto flat. Place rack directly on top of baking sheet with tomatoes so juices will drip down.
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Place baking sheet and rack into oven. Remove prosciutto after 3-4 minutes, once crispy. Return tomatoes and bake until skins burst, around 25 minutes.
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Crumble prosciutto and set aside.
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In large skillet or Dutch oven over medium heat, heat olive oil; add parsley, oregano, red pepper flakes and garlic; cook until fragrant, around 5 minutes. Remove from heat and let cool slightly.
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Add mozzarella, pasta water and pasta. Mix while still warm so cheese melts slightly.
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Add tomatoes to pasta mixture and gently toss to combine. Let mixture cool for a few minutes.
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Stir in basil and drizzle with vinegar or lemon juice.
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Serve topped with crumbled prosciutto and green onion.