Separate the egg yolks from the whites and strain only the yolks through a fine strainer to remove the clear yolk sac. place the egg yolks in a plastic bag and reserve. Cut a very tiny hole right before cooking them in syrup.
Make a thin sugar syrup by mixing together the sugar with the water in a large, flat frying pan over medium-high heat. Bring to a boil and reduce it to medium-low. Add the vanilla extract and swirl pan in order to vanilla mix in vanilla. Let sugar dissolve completely, approximately 3 minutes. Instead of vanilla extract, a cinnamon stick or other spices can be used to flavor the syrup (boil them for 5 minutes); however, before starting to drop the yolks into the syrup, remove the cinnamon stick/spices.
Slowly drizzle the yolks into the sugar syrup in circles, dropping only a few strands at a time from a height of about 2 inches or 5 cm. The threads must be long [at least 5 inches (13 cm) or more]. When they have fully cooked (about 1 minute per batch), they turn a bright yellow-gold. Remove them from the sugar syrup with a slotted spoon and placed into medium to large sized sieve. Hand-sprinkle ice cold water over them and let drain. Then, untangle them from one another with two long toothpicks. If covered with wax paper in an airtight container, the fio de ovos or eggs threads will keep for up 3 days in the fridge or 15 days in the freezer.
Fios de ovos can be used as a topping, an accompaniment, or decorative garnish for either savory or sweet dishes.
NOTE: Alternatively, an empty and cleaned-out condensed milk can may be used to drop the yolks into the syrup instead of the plastic bag with a tiny hole. In that case, make 5 holes in the bottom of the can with a thin nail. Another thing: Fill the can by 1/3 full at a time to avoid making too many threads at once.