Cook orecchiette in water with the tablespoon salt until 1 minute short of the package time. Reserve 1 cup pasta water before draining.
Meanwhile, season cutlets with 1/4 teaspoon salt and pepper. Heat oil in a large skillet and cook for 4 to 5 minutes per side until at least 165°F; transfer to a board to rest.
Melt butter in the skillet over medium heat and cook shallot for 2 minutes. Add spinach in handfuls and cook for 2 to 3 minutes until wilted and excess moisture evaporates.
Stir flour into the spinach for 1 minute, then gradually add wine and milk. Simmer gently for 3 minutes and add nutmeg and remaining salt.
Slice chicken and return with the pasta and 1/4 cup reserved pasta water. Toss over low heat for 1 to 2 minutes, stir in Parmesan off the heat and add more water if needed before serving.