Combine the butter, water and ½ teaspoon salt in a medium saucepan; bring to a simmer over medium-high heat. Reduce heat to medium; add flour all at once, and stir quickly with a wooden spoon until the dough pulls away from the sides of the pan and the bottom of the pan is clean. The dough should be glossy and smooth but still moist.
Continue stirring 5 minutes longer or until a thin coating forms on the bottom and sides of the pan, adjusting the heat as necessary to prevent the dough from coloring. Transfer dough to stand mixer bowl; stir in squash, herbs and remaining ½ tablespoon salt until combined. Stir in cheese. With mixer on low speed, add 2 eggs, one at a time, beating until each egg is completely incorporated before adding the next. Increase speed to medium; add 1 egg, mix until fully incorporated. Lift some of the dough with a rubber spatula; turn spatula, letting the dough fall off. The dough should move down the spatula very slowly; if it doesn't move at all, or the dough seems dry and just falls off in 1 piece, beat in an additional egg until incorporated.
Transfer dough to large pastry bag; let stand 30 minutes at room temperature. Bring a large saucepot of lightly salted water to a simmer. Line a sheet tray with paper towels; line second sheet tray with parchment paper.
Squeeze pastry bag over pot of simmering water with one hand while using scissors to cut off 1-inch pieces of dough with the other hand; cook about 30 gnocchi per batch. Water should remain at a simmer but not boil. Once gnocchi float to the top, cook 1-2 minutes or until gnocchi are still slightly undercooked in the center; transfer to paper towel-lined tray with slotted spoon. Repeat with remaining dough. Transfer drained gnocchi in single layer to parchment-lined tray; cover with plastic wrap and refrigerate at least 30 minutes. Yield: about 110 gnocchi.
For the Roasted Chicken
Refrigerate chickens, uncovered for 24 hours to dry out skin; remove chickens and let stand at room temperature 1 hour. Pat chickens dry with paper towel. To remove each wishbone, locate the wishbone with your fingers in neck/breast area; use a paring knife to gently pierce flesh above wishbone. Run the knife along upper side and then lower side of the wishbone; gently pull out wishbone with fingers.
Sprinkle each chicken cavity with 1 teaspoon salt and ¼ teaspoon pepper; add garlic, leek greens and herbs. Brush chickens with oil; sprinkle each with 2 teaspoons salt and ¾ teaspoons pepper. Truss the chickens tightly with butcher’s twine. Place chickens on wire rack on sheet tray. Roast chickens at 475°F for 25 minutes; reduce temperature to 400°F and roast 45 minutes longer or until chickens reach 160°F in thickest part of thigh and juices run clear. Tent chickens with aluminum foil; let stand 20 minutes. Carve chickens and sprinkle with fleur de sel. Yield: 8 servings.
For the Corn Cream
Juice corn kernels with a juicer or blender; strain with fine-mesh strainer into small saucepan. Heat juice over medium-low heat 20 minutes or until it thickens and coats the back of a spoon, stir frequently. Stir in butter; strain through fine-mesh strainer into bowl; stir in salt and white pepper. Yield: 2½ cups.
For the Mushrooms
Heat oil and butter in large sauté pan over medium-high heat until hot. Add mushrooms, salt and pepper; sear for 3-4 minutes or until tender, turning once. Transfer mushrooms to paper towel-lined plate.
For the Pancetta
Heat oil in large skillet over medium heat. Add pancetta and cook 15 minutes or until crisp, stirring occasionally; transfer to paper towel-lined plate. Yield: ¾ cup.
For the Crispy Sage
Heat oil in large sauté pan over medium-high heat until hot; add sage and cook 30-45 seconds or until crisp, turning once. Transfer sage to paper towel-lined plate; sprinkle with salt.
Reheat 1 portion of chicken with some chicken stock in oven; finish under broiler. Heat 1 tablespoon butter in sauté pan over medium heat until hot and slightly browned; add ¾ cup gnocchi and cook 4 minutes or until evenly browned. Add 6 mushrooms and 1½ tablespoons pancetta; cook just until heated through.
Spoon 2 ounces corn cream onto plate; place chicken and gnocchi mixture over corn cream. Garnish with fried sage, shaved Parmesan and chives.