Wash and pat dry the zucchinis, then finely grate them. Peel and mince the garlic.

Heat the olive oil in a skillet over medium heat. Sauté the grated zucchini and garlic for about 5 minutes. Add the fresh goat cheese and dried basil. Season with salt and pepper, then cook for a few more minutes until the mixture is smooth and well combined. Set aside about 3 tablespoons of the mixture for the sauce.

Stuff the cannelloni with the zucchini and goat cheese mixture using a small spoon. Arrange them in a buttered baking dish.

In a bowl, mix the reserved filling with the heavy cream and egg yolk. Season with salt and pepper to taste. Pour the mixture over the cannelloni to coat them evenly.

Sprinkle the grated Parmesan over the top. Bake in a preheated oven at 425°F (220°C) for about 10 minutes, or until the top is golden and bubbly.

Serve immediately with a fresh green salad for a light and complete meal.