Make the hollandaise: Put the egg yolks, mustard, cayenne, and water into a blender. Blend this together and then slowly start pouring in the warm butter with the blender running. Stop the machine when all the butter has been incorporated. Taste and adjust the seasoning with lemon juice, salt, and pepper. Thin the hollandaise with a little water if it is too thick. Keep warm while you fry the hash and poach the eggs.
Fry the hash: Heat the oil and butter in a nonstick skillet over medium heat, add the onion, and cook until soft, about 5 minutes. Add the corned beef, vegetables,and chives; cook until the bottom is brown and crisp, about 5 to 10 minutes.
Poach the eggs: Meanwhile, fill a large skillet with water, bring it to a boil, and add the vinegar and a large pinch of salt. Reduce the heat until the water is just barely bubbling. Crack each egg into a small bowl and carefully slide the egg into the water. Cook until the eggs are set, about 3 1/2 to 4 minutes. Remove the eggs with a slotted spoon and dab the bottom with paper towels to dry them off.
Serve: Place 1/4 of the hash onto a warm plate or shallow bowl. Top with a poached egg and a couple of tablespoons of hollandaise; garnish with chives.
Herbed Root Vegetables:
Put the olive oil and butter into a large pot over medium-high heat. Add the vegetables and toss to coat them well with the fat; season with salt and pepper. Add 1 cup water and bring to a boil. Lower the heat to a simmer, cover the pot, and cook until the vegetables are tender, about 20 minutes.
Meanwhile, make the Herb Butter by combining the soft butter and herbs together; season with some salt and pepper.
To serve, spread some Herb Butter in the bottom of a bowl. Add the hot vegetables and dot with more Herb Butter. Moisten with some of the cooking liquid and serve.