Well-stocked supermarkets carry mirin (rice wine) and dried udon (chewy wheat noodles) in the ethnic food aisle; lo mein noodles are a suitable sub.
In a 10-inch skillet combine 4 cups water and 1 Tbsp. vinegar. Bring to boiling; reduce heat to simmering (bubbles should begin to break surface of water). Break 1 egg into a cup and slip egg into simmering water. Repeat with 3 additional eggs, allowing each egg an equal amount of space in the water. Simmer 3 to 5 minutes or until whites are completely set and yolks begin to thicken but are not hard. Remove eggs. Season to taste with salt and pepper.TO MAKE AHEAD: Poach eggs as directed. Place cooked eggs in a bowl of cold water. Cover and chill up to 2 hours. To reheat eggs, in a saucepan bring water to simmering. Using a slotted spoon, slip eggs into the water; heat 2 minutes. Remove with slotted spoon.