This is my ode to one of America's greatest sandwiches, the Reuben--corned beef, Swiss cheese, sauerkraut, and Russian or Thousand Island dressing on rye. I top my open-face version with a poached egg and a cheater's Thousand Island-style sauce that can be drizzled on as is, which is my preference, or heated gently in a small saucepan or the microwave. Even without the poached egg this is definitely a knife-and-fork toast.
On a large baking sheet, toast the bread for about 5 minutes, until lightly browned on the bottom.
Fill a large deep skillet about halfway with water and heat over medium heat until tiny bubbles appear on the bottom.
In a medium bowl, whisk together the mayonnaise, ketchup, relish, paprika, Worcestershire sauce and lemon juice. Season the sauce generously with salt and pepper.
Stir the vinegar and a generous pinch of salt into the hot water in the skillet. One at a time, crack the eggs into a small bowl and very carefully slide them into the water. Poach over medium heat until the whites are just firm and the yolks are runny, 3 to 5 minutes. Using a slotted spoon, transfer the eggs to a plate.
Meanwhile, top each toast with a slice and a half of cheese, one-quarter of the corned beef and 1/4 cup of the sauerkraut. Bake for about 5 minutes more, until the cheese is just melted. Top each toast with a poached egg and drizzle with the Reuben sauce. Sprinkle with snipped chives and serve immediately.