Melt butter in a large nonstick saute pan over medium-high heat. Add gnocchi and cook until golden brown, tossing the gnocchi occasionally. Add parmesan, stirring to coat. Saute for another minute or until cheese melts. Season with black pepper and remove from heat. Transfer gnocchi to a plate and set aside.

In the same pan, heat olive oil over medium heat. Add shallot and garlic and saute for 1-2 minutes or until shallots are softened. Add Calabrian chilies, dried basil, and tomato paste, stirring to combine. Suate for 2 minutes or until tomato paste is slightly darkened in color.

Add vodka and bring to a boil, stirring constantly. When all of the liquid has evaporated, add half and half. Bring to a simmer, stirring to combine. Season sauce with salt and pepper.

Add gnocchi back to the pan and add ¼ cup water. If the sauce is too thick, add additional water. Adjust seasoning once more. Transfer gnocchi to plates and top with extra grated parmesan.