Trim the ends from the zucchini.
Using a mandoline or a sharp vegetable peeler, cut the zucchini lengthwise into paper-thin slices.
Arrange 6 to 9 slices on each of four plates, either overlapping in rows or in a basket-weave pattern to cover the surface.
In a small bowl, whisk together 3 tablespoons (45 ml / 1.5 fl oz) of the oil, 1 tablespoon (15 ml / 0.5 fl oz) of the lemon juice, 1 teaspoon of the zest, ¼ teaspoon of the salt, the sugar, and a pinch of pepper.
Drizzle this vinaigrette over the zucchini on each plate.
Set aside to marinate for at least 15 minutes, which will soften and flavor the squash.
Preheat the oven to 425°F / 220°C.
Clean the king oyster mushrooms by wiping them with a damp cloth. Trim the ends and slice the mushrooms into ½-inch / 1. 25 cm thick rounds.
In a mixing bowl, combine the mushroom slices with the remaining 2 tablespoons (30 ml / 1 fl oz) oil, the white wine, 1 teaspoon of the salt, and the ½ teaspoon pepper.
Toss gently to coat and set aside.
In a separate small bowl, mash together the softened butter, minced garlic, fresh herbs, red pepper flakes, the remaining 1 teaspoon zest, remaining 1 tablespoon (15 ml / 0.5 fl oz) lemon juice, the egg yolk, panko, remaining ¼ teaspoon salt, and the final pinch of pepper until well combined.
Arrange the marinated mushroom slices in a single layer in a baking dish.
Top each mushroom round with a small dollop of the garlic-herb butter mixture.
Bake for 10 to 12 minutes, or until the mushrooms are tender and the panko topping is golden brown and bubbling.
To serve, carefully arrange 4 to 5 of the hot mushroom scampi rounds atop each plate of zucchini carpaccio.
Serve immediately.