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1
If using fresh peas, parboil them in a small saucepan of boiling salted water until softened, 3-5 minutes; drain and set aside.
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2
Trim the tough ends and wilted spots from the mushrooms.
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3
Wipe them clean with a damp paper towel; slice them thin and set aside.
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4
Bring 6 quarts of salted water to a boil in an 8-quart pot over high heat.
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5
Heat the olive oil in a large skillet over medium heat.
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6
Whack the garlic cloves with the flat side of a knife and add them along with the prosciutto to the oil.
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7
Cook, stirring, until lightly browned, about 4 minutes.
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8
Stir in the mushrooms, season them lightly with salt and pepper, , and cook, stirring, until they are lightly browned and wilted, about 7 minutes.
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9
Pour in the tomatoes, season them lightly with salt and pepper, and bring the sauce to a boil.
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10
Lower heat so the sauce is at a lively simmer and cook 5 minutes.
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11
Stir in the peas and chopped parsley into the sauce and cook until the peas are tender, about 3 minutes.
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12
While the sauce is simmering, stir the bucatini into the boiling water; return to a boil, stirring frequently.
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13
Cook the pasta, semi-covered, stirring occasionally, until done, about 10 minutes.
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14
Drain the pasta, return it to the pot, and pour in about 3/4 of the sauce.
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15
Bring the sauce and pasta to a boil, tossing to coat the pasta with sauce.
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16
Check the seasoning, adding salt and pepper if needed.
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17
Remove the pot from the heat and stir in the grated cheese.
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18
Transfer the pasta to a warm platter, top with the remaining sauce, and serve immediately.