Heat the olive oil (or reserved sun-dried tomato oil) in a large braiser or skillet over medium-high heat.
Add the diced chicken, season with 1/2 teaspoon salt and 1/2 teaspoon black pepper, and cook until the pieces start to brown, about 2–3 minutes.
Add the chopped shallots, minced garlic, and chopped sun-dried tomatoes; stir and cook until the shallots soften, about 1–2 minutes.
Pour in the half and half and chicken broth, stir to combine, and bring the mixture to a simmer.
Add the uncooked gnocchi to the simmering sauce, stir to submerge, and cook until the gnocchi are tender and the sauce has thickened, about 4–5 minutes.
Fold in the chopped spinach and cook until wilted, then taste and adjust seasoning with additional salt and pepper if needed.
Stir in the shredded mozzarella and grated parmesan until the cheeses melt and the sauce becomes creamy, then serve warm.