For the gremolata, combine lemon zest on a cutting board with anchovy, parsley and garlic. Reserve.
Bring a pot of water to a boil for the pasta. Once boiling, salt water and cook pasta to al dente.
Meanwhile, season chicken with salt and pepper on both sides then dredge in flour to coat. Beat eggs in a shallow dish with splash of water. Dip both sides of the chicken in the egg.
Heat 3 tablespoons EVOO and 2 tablespoons butter in a large skillet over medium to medium-high heat. Cook chicken in 2 batches; 2 minutes on each side. Remove to a warm platter and reserve.