Heat 1 tablespoon olive oil in a large high-sided skillet or Dutch oven over high heat.
Add the diced chicken and cook undisturbed for about 2 minutes to brown the pieces, stirring as needed to cook through.
Reduce heat to low, add 1 tablespoon butter and the minced garlic, and cook, stirring, for about 30 seconds until fragrant.
Stir in 1/4 cup pesto until combined.
Pour in 1 cup heavy cream and 1/4 cup chicken broth, stir to combine, and bring the mixture to a gentle simmer.
Add 1.1 pounds uncooked potato gnocchi, stir to distribute, cover the pan, and cook over medium-low heat for 5 minutes.
Uncover, stir well, and cook uncovered for 2 more minutes until the sauce has thickened and the gnocchi are tender.
Remove from heat, stir in 1/2 cup grated Parmesan and 1/4 cup basil leaves until the cheese melts and basil wilts.
Taste and season with salt and black pepper as needed, then serve immediately.
