In a large oven-safe skillet over medium-high heat, heat the butter and olive oil until the butter is melted and lightly bubbling. Add the gnocchi and cook, stirring frequently, for about 5-7 minutes, or until the gnocchi is softened and golden-brown on all sides.
Add the zucchini. Season with a pinch of salt and pepper, cook for a few more minutes, until the zucchini is bright green and beginning to soften. Add the garlic and cook for an additional minute, until fragrant.
Remove the pan from the heat. Stir in the chicken, pesto, and half-and-half until evenly combined. Sprinkle the mozzarella and Parmesan cheese evenly over the top.
Transfer the pan to the preheated oven. Bake for 15-20 minutes, or until the cheese is melted and bubbling. Let cool for a few minutes, then scoop into bowls and serve topped with a few fresh basil leaves.