Preheat oven to 375ºF; grease a 9-by-13-inch casserole dish. Bring a pot of salted water to a boil. Add pasta; cook for 7 minutes. Add broccoli; cook until broccoli is tender, 3 to 4 minutes longer. Drain, rinse with cold water and drain again. Pour into casserole.
In a saucepan over medium heat, melt butter. Remove 1 Tbsp. from pan and drizzle over panko in a small bowl. Add flour to pan; whisk until smooth and bubbling, 1 minute. Slowly whisk in milk. Add paprika and nutmeg; season with salt and pepper. Stir until sauce begins to boil, about 8 minutes. Remove from heat.
Stir 1 1/2 cups Cheddar and all of Gruyère and Parmesan into sauce until smooth. Stir in ham.
Stir sauce into pasta mixture. Sprinkle with remaining Cheddar and reserved panko. Bake until top browns and sauce bubbles, about 30 minutes.