Prepare the pesto (homemade or jarred) as needed.
Cut the chicken into 1-inch wide by 2-inch long strips.
In a bowl, mix egg whites, lemon juice, oregano, and 2 tablespoons of pesto. Stir to combine. Place the chicken strips in a quart-sized zipper seal bag with the marinade and let them marinate for 30 minutes.
In a small frying pan, toast the pine nuts over medium heat for about 5 minutes until lightly browned. Do not overcook.
In a separate bowl, mix the flour, salt, and pepper. Dip the marinated chicken strips into the flour mixture, coating both sides.
Heat 2 tablespoons of butter and 2 tablespoons of olive oil in a 14-inch sauté pan over medium-high heat until hot and frothy. Add the chicken strips in a single layer and cook for 2-3 minutes per side, or until cooked through. You may need to cook in 2-3 batches, adding more butter and oil as necessary. Remove the cooked chicken to a platter and set aside.
Cook the linguini according to the package instructions.
In the same pan, sauté the shallots over medium heat for 1-2 minutes until translucent. Add the garlic and cook for 30 seconds. Add the prosciutto, parsley, and toasted pine nuts. Cook for 1 more minute.
Add the cream and 3 tablespoons of pesto to the pan. Stir in salt and pepper to taste, then mix in 1/3 cup of Parmesan cheese until melted into the sauce. Reserve the rest of the Parmesan for serving.
Add the cooked chicken to the sauce and gently stir to combine.
To serve, mix 2-3 tablespoons of pesto with the cooked linguini and place it on a serving platter. Top with the chicken and cream mixture. Sprinkle the remaining Parmesan cheese and top with arugula. Alternatively, serve individual portions with Parmesan and arugula on top.