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1
Cut 8 of the zucchini crosswise into 2-inch pieces.
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2
Using a melon baller, hollow out each piece, leaving a 1/4-inch shell.
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3
Mince the scooped-out flesh along with the 4 remaining zucchini.
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4
In a large, deep skillet, melt the butter in the oil.
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5
Add the shallots and cook over moderate heat until translucent, about 3 minutes.
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6
Add the chopped zucchini and cook, stirring occasionally, until tender and lightly browned, about 25 minutes.
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7
Stir in the parsley and thyme.
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8
Transfer the zucchini to a food processor and puree.
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9
Scrape the puree into a bowl, stir in the Parmigiano-Reggiano and season with salt and pepper.
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10
Let the filling cool, then stir in the eggs.
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11
Preheat the oven to 350.
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12
Generously butter a large baking dish.
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13
Set the zucchini cups in the baking dish, hollowed side up.
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14
Spoon the filling into the cups, mounding it slightly.
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15
Bake the stuffed zucchini for about 1 hour and 10 minutes, or until tender and lightly browned.
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16
In a food processor, pulse the egg yolks with the olive oil until a smooth paste forms.
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17
Add the pecorino and the 1/3 cup of water and process until the sauce is smooth and has the consistency of heavy cream; add a little more water if the sauce seems thick.
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18
Season with salt.
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19
Spoon the room-temperature pecorino sauce onto serving plates and arrange about 3 warm zucchini cups on top.
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20
Serve at once.