Bring a large pot of salted water to a boil for the spaghetti. Steam broccoli until bright green and tender. While they cook, pound the chicken into even, thin cutlets.
In a large skillet, melt the butter. Sprinkle in the flour and whisk for 1 minute until fully blended. Slowly add the milk while whisking to form a smooth, thickened base.
Whisk in garlic powder, onion powder, and mustard powder. Add Parmesan and Neufchâtel, stirring until melted. Fold in shredded Cheddar in batches until smooth and velvety. Keep warm on low heat.
Combine Italian breadcrumbs, chopped parsley, and grated Parmesan in a shallow dish.
Dip each chicken cutlet in beaten egg, then press firmly into the breadcrumb mixture to fully coat both sides.
Heat oil to 350°F (175°C). Fry chicken for about 5 minutes per side until crispy, golden, and cooked through. Transfer to a cooling rack.
Drain spaghetti, reserving 1 cup of starchy pasta water. Add spaghetti and broccoli to the cheese sauce, stirring to coat. Thin with pasta water as needed for silky texture.
Slice chicken cutlets and place them on top of the cheesy broccoli spaghetti. Garnish with fresh parsley and Parmesan.