께
Curiosity
chingu-ege seonmul-eul jwoyo ("I give a gift to a friend") sounds fine. But if the receiver is your teacher, can you still use ege?
Intuition
Honor extends not only to the subject but also to the person the action reaches (the object). kke marks deference toward the receiver: "the one this action reaches ranks high."
Visualization
Step up one level from a plain dative to an honored one: chingu-ege → seonsaengnim-kke. The ege/hante turns into kke.
Essence
kke is object honorification and often pairs with deurida, the honorific of "to give": seonsaengnim-kke seonmul-eul deuryeoyo ("I give a gift to the teacher"). ege is written/formal, hante is colloquial — both become kke for an honored receiver. Do not confuse it with kkeseo (subject), which looks similar.
Examples
Mini-quiz
"seonsaengnim__ seonmul-eul deuryeoyo." (I give a gift to the teacher) — which fits an honored receiver?