In a large pot of boiling water, cook the fettuccine until tender. Drain and reserve 1/3 cup cooking water.

Skin chicken breasts and cut into 1 inch cubes. On a sheet of waxed paper, combine two tablespoons of the flour, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Dredge the chicken in the flour mixture shaking off the excess.

In a large nonstick skillet, heat the oil until hot but not smoking over medium heat. Add the chicken and cook, stirring, until lightly browned, about 2 minutes. Add the remaining 1 tablespoon flour and stir until the chicken is coated. Gradually add the milk, stirring until smooth. Add the cream cheese, reserved 1/3 cup pasta water, the nutmeg, remaining salt and remaining pepper. Bring to a boil, reduce to a simmer, cover, and cook until the sauce is slightly thickened and the chicken is cooked through, about 5 minutes.

Add the peas and cook, uncovered, just until the peas are warmed through, about 1 minute. Transfer to a large bowl, add the fettuccine and Parmesan and toss to combine. Spoon the fettuccine mixture onto 4 plates, sprinkle the scallions over, and serve.