Make the hollandaise sauce: Melt the butter in a saucepan over low heat. Remove the pan from the heat and let stand 5 minutes. Skim the foam from the top and discard. Slowly pour the clear butter into a liquid measuring cup, discarding the milk solids in the bottom of pan — you should have 1/2 cup clarified butter.
In a metal bowl (or the top of a double boiler), combine the egg yolks with the cool water. Whisk until frothy, 1 to 2 minutes. Place the bowl over a pot filled with a few inches of simmering water (do not let the bottom of the bowl touch the water) and cook, whisking, until the eggs are thickened, 5 to 8 minutes. Don’t rush this step, because the sauce needs to be based on fluffy cooked eggs. Keep the heat low as well to avoid overheating and scrambling the eggs.
Remove the bowl from the pot and place it on top of a kitchen towel to keep it from tipping over. Little by little, whisk in 1/2 cup clarified butter. When all the butter is added, whisk in the lemon zest and juice and a pinch each of kosher salt and pepper.
Make the eggs Benedict: Place the English muffin halves on a flat surface and, one by one, roll over them with a rolling pin so they flatten and expand. Toast them lightly in a toaster oven.
Fill a large wide pot with about 3 inches of water; bring to a simmer. Add the vinegar (this will help the eggs keep their shape). Crack each egg into a separate small bowl. Gently drop the eggs, one by one and at some distance from one another, into the water. Lower the heat and poach the eggs until the whites are firm and completely white, 2 to 3 minutes. Use a slotted spoon to remove them to a kitchen towel to drain. Season with kosher salt and pepper.
Season the avocado slices with kosher salt; place on each of the English muffin halves. Top with a poached egg, some of the hollandaise sauce, Cajun seasoning, parsley and sea salt.