In a small bowl, whisk together the extra-virgin olive oil, balsamic glaze, minced garlic, salt, and pepper. Let sit at room temperature while you cook the gnocchi so the garlic mellows.

Bring a large pot of salted water to a boil. Add gnocchi and cook just until they float, about 2-3 minutes. Drain well and pat dry with paper towels - this is crucial for crisping. Heat 1 tablespoon olive oil and the butter in a large nonstick skillet over medium-high heat. Add gnocchi in a single layer and cook undisturbed until golden and crisp on one side, 3-4 minutes. Toss and continue cooking until most sides are browned and crispy, about 6-8 minutes total.

While gnocchi are still warm, transfer them to a large bowl. Add halved cherry tomatoes and mozzarella pearls. Pour the dressing over and toss gently to combine - the warm gnocchi will slightly soften the mozzarella in the best way.

Tear the basil leaves over the salad and give one final gentle toss. Taste and adjust salt if needed. Serve immediately while the gnocchi are still warm against the cool tomatoes and cheese.