Heat a large pot of salted water to a boil. Add pasta and cook until al dente. Drain and set aside.
Meanwhile, steam broccoli and set aside. I use my rice cooker. Simply put the florets in the cooker with 2-3 tbs of hot water and turn the cooker on. Broccoli will be bright green and tender-crips when done. Don't forget about it in there - it'll finish cooking before the rice cooker clicks off, so if you forget about it, you'll overcook your broccoli. Mine takes less than 10 minutes to cook.
Make sure the carrots and potatoes are chopped very small; this will reduce the cooking time greatly. Place the chopped potatoes and carrots in small saucepan that had a lid. Add the 1 cup of water. Boil covered until tender, 10-15 minutes.
While the potatoes and carrots are cooking, add the remaining ingredients to the blender or food processor (cheddar cheese, soy sauce, lemon juice, mustard, parmesan, margarine, salt, pepper). Once potatoes and carrots are done cooking, add them and their cooking water to the blender as well. Blend until VERY smooth. If needed, add milk or water 1 tbs at a time to thin. Taste for seasoning.
Toast the bread and process into crumbs.
Toss the cooked pasta and broccoli with the sauce. Place in a casserole dish and top with breadcrumbs, black pepper, and paprika.
Bake for 25 minutes at 350 F until browned and bubbly. Serve and enjoy!