Pat chicken dry and season with half the salt and the black pepper.
Heat olive oil and butter in a large skillet over medium-high. Sear chicken until lightly browned, 4–5 minutes. Transfer to a plate.
Lower heat to medium. Sauté garlic and red pepper flakes in the skillet until fragrant, about 30 seconds.
Pour in chicken broth and heavy cream. Scrape up browned bits and bring to a gentle simmer.
Stir in gnocchi. Cover and cook, stirring once or twice, until tender and slightly thickened, 4–5 minutes.
Return chicken and any juices to the pan. Fold in pesto and tomatoes. Simmer 2 minutes to warm through.
Add spinach and the remaining salt. Stir until wilted. Mix in Parmesan and lemon juice. Taste and adjust seasoning.
Serve hot, topped with extra Parmesan, toasted pine nuts, and fresh basil.