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