Pat the chicken pieces completely dry with paper towels. This is essential for browning. Season with 1/2 teaspoon salt and the black pepper. Heat the olive oil in a large skillet over medium-high heat until shimmering. Add the chicken in a single layer without crowding (work in batches if needed). Cook undisturbed for 4-5 minutes until deeply golden on the first side, then flip and cook 3 minutes more. The chicken does not need to be fully cooked through yet. Transfer to a plate.

Reduce heat to medium. Add the garlic to the same skillet and cook for 30 seconds until fragrant but not browned. Pour in the chicken broth, scraping up all the browned bits from the bottom of the pan with a wooden spoon. These bits are pure flavor. Let the broth simmer for 2 minutes to reduce slightly.

Stir in the heavy cream and remaining 1/4 teaspoon salt. Add the gnocchi in an even layer, then nestle the seared chicken and any accumulated juices on top. Bring to a gentle simmer, cover, and cook for 5-6 minutes. The gnocchi will soften and absorb the sauce. Uncover and stir gently - the gnocchi should be tender but not mushy when pierced with a fork.

Remove from heat. Stir in the pesto, Parmesan, and lemon juice until the sauce is creamy and coats everything evenly. The residual heat will melt the cheese perfectly. Taste and adjust seasoning if needed. Let rest for 2 minutes so the sauce thickens slightly. Serve in shallow bowls topped with torn fresh basil and extra Parmesan.