This is the 3532nd most frequent Japanese word.
茹で
Boil (typically food in water).
In this sentence, '茹でる' functions as a verb, expressing the action of boiling vegetables.
私は野菜を茹でるのが好きです。
I like to boil vegetables.
Here, '茹でた' is used as an adjective derived from '茹でる', describing the state of the eggs after being boiled.
もう茹でた卵を冷蔵庫に入れましたか。
Did you already put the boiled eggs into the refrigerator?
In this instance, '茹でたて' combines '茹でる' with 'たて', emphasizing items that are freshly boiled.