Learn Hardcore Korean: choreul kkeugo naseo keikeu jogageul nanwo meogeoyo. - After extinguishing the candles, share and eat the cake slices.

-고 나서 attaches to a verb stem to express “after doing …”. It indicates that the first action is fully completed before the second begins.
• 초를 끄고 나서 = “after blowing out the candle(s)”
In contrast, -고 alone merely links two actions (like “and”) without stressing the sequence.

Could I replace -고 나서 with -은/ㄴ 후에? If so, what’s the nuance difference?

Yes. You could say:
초를 끈 후에 케이크 조각을 나눠 먹어요.
Here 끈 is the modifier form of 끄다 (past attributive).
Nuance:
• -고 나서 – more conversational, everyday speech.
• -은/ㄴ 후에 – slightly more formal or written style.

Why is it 케이크 조각을 instead of 조각 케이크? Which is the head noun?

In 케이크 조각, 조각 (“piece”) is the head noun and 케이크 (“cake”) is its modifier: “a piece of cake.” We add -을 because 케이크 조각 is the direct object of 나눠 먹어요. Saying 조각 케이크 would sound like a fixed compound meaning a “slice‐style cake” (e.g., individually sold slices), not the act of sharing pieces.

Why is it 나눠 먹어요 instead of just 나눠요 or 먹어요?

나눠 먹다 is a compound verb meaning “divide/share and eat.”

나누다 → contracted to 나눠
- 먹다 → 나눠 먹다
  Thus you need the second verb’s ending (-어요 on 먹다) to complete the meaning.

What’s the difference between 나눠 먹다 and 같이 먹다?

• 나눠 먹다 – emphasizes dividing something into portions and eating your share.
• 같이 먹다 – emphasizes eating together (you might each have your own portion, but you share the time).

Why do Koreans say 초를 끄다 for “blow out a candle” instead of 불을 끄다?

• 초 specifically means “candle,” so 초를 끄다 is “extinguish a candle.”
• 불을 끄다 is used for turning off general flames or lights (e.g., stove, lamp). You could say 초의 불을 끄다, but it’s more natural to say 초를 끄다.

Why is the verb ending -어요 used here? Can I use -습니다 instead?

-어요 is the polite (해요체) ending for friendly or informal‐polite speech. If you want a more formal tone (하십시오체), you’d use -습니다:
초를 끄고 나서 케이크 조각을 나눠 먹습니다.