How to Make It
Cut 8 of the zucchini crosswise into 2-inch pieces. Using a melon baller, hollow out each piece, leaving a 1/4-inch shell. Mince the scooped-out flesh along with the 4 remaining zucchini.
In a large, deep skillet, melt the butter in the oil. Add the shallots and cook over moderate heat until translucent, about 3 minutes. Add the chopped zucchini and cook, stirring occasionally, until tender and lightly browned, about 25 minutes. Stir in the parsley and thyme.
Transfer the zucchini to a food processor and puree. Scrape the puree into a bowl, stir in the Parmigiano-Reggiano and season with salt and pepper. Let the filling cool, then stir in the eggs.
Preheat the oven to 350°. Generously butter a large baking dish. Set the zucchini cups in the baking dish, hollowed side up. Spoon the filling into the cups, mounding it slightly. Bake the stuffed zucchini for about 1 hour and 10 minutes, or until tender and lightly browned.
In a food processor, pulse the egg yolks with the olive oil until a smooth paste forms. 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. Season with salt.
Spoon the room-temperature pecorino sauce onto serving plates and arrange about 3 warm zucchini cups on top. Serve at once.