Heat the oil in a large braiser or skillet over medium-high heat until shimmering.
Add the cut chicken, season with 1/2 teaspoon salt and 1/2 teaspoon black pepper, and cook, stirring occasionally, until the pieces start to brown, about 2–3 minutes.
Add the chopped shallots, minced garlic, and chopped sun-dried tomatoes and cook, stirring, until the shallots soften and garlic is fragrant, about 1–2 minutes.
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, cover if needed to maintain a simmer, and cook until the gnocchi are tender and the sauce has thickened, about 4–5 minutes; stir occasionally so gnocchi cook evenly and absorb sauce.
Add the chopped spinach and fold it into the hot mixture until wilted, then taste and adjust seasoning with salt and pepper if needed.
Remove from heat and stir in the shredded mozzarella and grated Parmesan until the cheeses melt and the sauce is creamy; serve immediately.