Prepare the gnocchi according to the package directions and 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 softened shallots and garlic to a food processor or blender with the drained roasted red peppers and the broth; blend 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 and heat until warmed through.
Add the baby spinach and cook, stirring, until wilted, about 4–5 minutes.
Toss the cooked gnocchi and sliced grilled chicken into the skillet with the sauce and stir to combine and heat through.
Stir in the chopped fresh basil, divide among bowls, and serve with extra Pecorino if desired.