Heat the olive oil or reserved sun-dried tomato oil in a large braiser over medium-high heat until shimmering.
Add the chicken pieces, season with the salt and black pepper, and cook, stirring occasionally, until the chicken begins to brown, about 2 to 3 minutes.
Add the chopped shallots, minced garlic, and chopped sun-dried tomatoes and cook, stirring, for 1 to 2 minutes until fragrant.
Pour in the half and half and chicken broth, stir to combine, and bring the mixture to a gentle simmer.
Stir in the uncooked gnocchi, then cook, stirring occasionally, until the gnocchi are tender and the sauce has thickened, about 4 to 5 minutes.
Add the chopped spinach and fold it into the sauce until wilted, then taste and adjust seasoning with salt and pepper if needed.
Stir in the shredded mozzarella and grated Parmesan until the cheeses melt and the sauce is creamy, then remove from heat and serve.