Boil pasta and broccoli: Bring a pot of water to a boil. Add pasta and cook until tender. Add frozen broccoli florets for the last 2 minutes. Drain together.
Cook chicken: Season chicken breast with salt and pepper. Heat oil in a skillet over medium heat. Cook chicken for 5-7 minutes per side until golden brown and cooked through. Let rest, then chop into cubes.
Start the sauce base: Reduce skillet heat to medium-low. Add butter and melt. Whisk in browned bits from the bottom.
Add liquids and spices: Pour in evaporated milk, whisking to dissolve browned bits. Whisk in garlic powder, smoked paprika, and salt.
Melt cheese: Gradually add shredded white cheddar, one handful at a time, whisking until fully melted before adding more. Keep heat low.
Combine: Add drained pasta, broccoli, and chopped chicken to the skillet with the cheese sauce. Stir until everything is coated and serve immediately.