Bring a large pot of water to a boil. Add the pasta and cook according to the timing on the package, about 8 minutes total. Add the chopped broccoli halfway through the cooking time so it becomes tender but not overcooked.
Drain the pasta and broccoli well and set aside.
Place the same pot back on the stove over low to medium heat. Add the butter, milk, salt, and pepper. Stir until the butter has fully melted.
Return the drained pasta and broccoli to the pot. Add the spreadable cheese wedge and shredded cheddar. Stir gently and allow the cheese to melt slowly, coating the pasta evenly. If the sauce seems too light, add a little more shredded cheese.
Mix in the chopped spinach and stir until it softens and blends into the warm pasta.
Serve right away. Adjust salt and pepper if needed.