Prepare the gnocchi according to the package directions; set aside once cooked and drained.
Heat 2 teaspoons olive oil in a large skillet over medium-high heat.
Add the minced shallots and garlic and sauté 2–3 minutes, until softened.
Transfer the shallots and garlic to a food processor or blender with the drained roasted red peppers and 1/2 cup broth; purée until smooth.
Pour the pepper purée back into the skillet and bring to a simmer over medium heat.
Stir in 1/4 cup half and half, 1/4 cup Pecorino Romano, 1/2 teaspoon kosher salt, and fresh black pepper; cook until heated through.
Add the baby spinach and cook 4–5 minutes, until wilted.
Combine the cooked gnocchi and sliced grilled chicken with the sauce in the skillet and toss to coat evenly.
Stir in the chopped fresh basil, divide among bowls, and serve with extra Pecorino if desired.