Prep the chicken: Pat dry and slice into thin cutlets or bite-size pieces. Season all over with salt, pepper, and Italian seasoning.
Cook the pasta: Bring a large pot of salted water to a boil.
Cook pasta until just shy of al dente. Reserve 1 cup of pasta water, then drain.
Sear the chicken: Heat olive oil in a large skillet over medium-high. Add chicken in a single layer and cook 3–4 minutes per side until golden and cooked through.
Transfer to a plate and tent with foil.
Build the aromatics: Lower heat to medium. Add butter to the skillet. Stir in shallot (if using) and cook 1–2 minutes until soft.
Add garlic and cook 30 seconds until fragrant.
Make the base: In a bowl, whisk milk with cornstarch until smooth. Pour into the skillet, stirring constantly. Cook 2–3 minutes until it gently thickens.
Add the creamy boosters: Lower the heat to medium-low.
Whisk in cottage cheese and Greek yogurt until smooth. Sprinkle in nutmeg and a pinch of red pepper flakes if you like heat.
Melt in the Parmesan: Add Parmesan a handful at a time, stirring until melted and glossy. If the sauce seems too thick, loosen with a splash of pasta water or chicken broth.
Taste and adjust salt and pepper.
Combine: Add chicken (and any juices) back to the skillet. Stir in spinach or broccoli if using. Fold in the cooked pasta, tossing to coat.
Add more pasta water as needed for a silky finish.
Finish and serve: Remove from heat. Squeeze in a touch of lemon for brightness and top with chopped parsley or basil. Serve hot with extra Parmesan on the side.