Bring a large pot of salted water to a boil and cook the gnocchi according to package directions until they float and are just tender, about 2–3 minutes; reserve 1/2 cup cooking water and use a slotted spoon to transfer gnocchi to a plate, patting lightly to remove excess moisture.

Heat 1 tablespoon olive oil and 1 tablespoon butter in a 12-inch skillet over medium-high heat; add gnocchi in a single layer and let sit 1–2 minutes to brown, stir and cook 30–60 seconds more, then transfer to a bowl and set aside (work in batches if needed).

Reduce heat to medium and add remaining 1 tablespoon butter to the skillet; sauté garlic and shallot until fragrant, about 1 minute, then add sun-dried tomatoes and stir 30 seconds.

Pour in 1/3 cup chicken broth or reserved gnocchi water, scraping up browned bits, then lower heat to medium-low and stir in 1 cup heavy cream and 1/4 cup grated Parmesan until smooth.

If the sauce is too thick, stir in up to 2 tablespoons reserved cooking water or broth until it reaches a saucy consistency; season with 1/2 teaspoon salt and 1/4 teaspoon pepper and taste.

Add spinach and stir until just wilted, about 1–2 minutes; return seared gnocchi to the skillet and gently fold to coat, warming through 1–2 minutes without a vigorous simmer.

Remove from heat and stir in 1 tablespoon lemon juice and 2 tablespoons chopped parsley; serve immediately with extra Parmesan.

If using rotisserie chicken, fold it in with the gnocchi to warm through; if using mushrooms, sauté them with the shallot before adding sun-dried tomatoes and proceed as directed.