Place rack in upper middle position of the oven and preheat the broiler to high.
Trim the woody ends of the asparagus off. In a saute pan (with lid), bring the water to a simmer. Add the woody asparagus ends plus 1/4 teaspoon of salt to the water and cook for 5 minutes. Using slotted spoon, remove and discard the woody asparagus ends.
Return water to a simmer. Add the asparagus to the water, cover and cook for 2 minutes, until just nearly tender (for very thin asparagus, you only need to cook 1 minute). With slotted spoon, remove the asparagus and reserve. Pour just 1/2 cup of the asparagus water into a measuring cup (discard the rest).
Wipe the saute pan dry and return to stove on medium heat. Melt the butter in the pan and with whisk or spoon, stir in the flour. Reduce heat to medium-low and cook until thickened and nutty smelling, about 3-5 minutes. Turn off the heat. Stir in most of the cheese (reserve a couple of tablespoons of cheese for the top of the asparagus) and the reserved asparagus water. Taste then season with salt and pepper.
Pour this thickened cheesy sauce into oven-safe casserole dish. Place the asparagus on top of the sauce. Sprinkle remaining cheese on top of asparagus. Broil for 6-8 minutes. Top with parsley. Serve immediately.