Heat 1 tablespoon olive oil in a large high-sided skillet or Dutch oven over high heat.
Add the diced chicken and cook for about 2 minutes, stirring occasionally, until the outsides start to brown.
Reduce the 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 with the butter and garlic.
Pour in 1 cup heavy cream and 1/4 cup chicken broth, stir to combine, and bring the mixture to a gentle simmer.
Add the 1.1 pounds uncooked potato gnocchi, cover the pan, and simmer over medium-low heat for 5 minutes to cook the gnocchi through.
Uncover, stir the gnocchi and sauce, and cook uncovered for 2 more minutes to thicken slightly.
Remove the pan from the heat, stir in 1/2 cup grated Parmesan and 1/4 cup basil leaves until the cheese melts and the basil is evenly distributed.
Season with salt and pepper to taste and serve immediately.