Combine the water and sugar together in a saucepan and bring to a boil. Reduce heat to medium-low; add the ginger slices and cook until fragrant, about 5 minutes.

Crack the eggs into individual bowls and gently drop them into the syrup. Cook the egg on one side until the white is partially set and opaque, about 3 1/2 minutes; gently turn over to cook the other side until egg white is opaque but not hard, and the yolk is still liquid, about 3 1/2 more minutes or to desired doneness. Spoon the eggs into individual bowls and spoon syrup over each egg. Garnish with the cooked ginger.