어떻게 V+ 어/아/여 드릴까요?

It is used to politely ask about what the listener wants when the speaker is doing a behavior for the listener. It is often used in a question sentence for offering service.

For ‘-어/아/여 드릴까요?’, if the stem of an action verb ends with ‘ㅏ, ㅗ’, you should use ‘-아 드릴까요?’, with another vowel(ㅓ, ㅜ, ㅡ, ㅣ) you use ‘-어 드릴까요?’. If it ends with ‘하-‘, you should use ‘-여 드릴까요(해 드릴까요?’.

앞머리는 어떻게 해 드릴까요? 짧게 잘라 드릴까요?
How would you like your bangs? Do you want me to cut it short?

다 쓴 통장은 어떻게 처리해 드릴까요? 여기 은행에서 폐기해 드릴 수 있습니다.
How would you like your used bank account processed? We can dispose of it here at the bank.

케이크는 어떻게 만들어 드릴까요? 원하는 디자인을 말씀해 주세요.
How would you like your cake made? Tell me the design you want.

Conversation:

우유를 좀 신청하려고요.
I’d like to order some milk.

네, 우유는 200밀리리터, 500밀리리터, 1리터 중에 어떤 걸로 하시겠어요?
Okay, among 200 milliliters, 500 milliliters, or 1 liter of milk, what would you want?

아이가 먹을 건데 유기농 우유도 있나요?
My child is going to drink it. Do you also have organic milk?

네, 아이들용으로 유기농 우유를 선택하시는 분이 많아요.
Yes, there are many people who choose organic milk for their children.

그럼 배달은 어떻게 해 드릴까요?
Then, how would you like your delivery done?

유기농 흰 우유 200밀리리터로 일주일에 세 번씩 한 달만 배달해 주세요.
Please deliver 200 milliliters of white milk three times a week only for a month.

6개월 계약하시면 1개월 무료로 연장해 드리는데 6개월 계약은 어떠세요?
If you sign a six-month contract, we offer you a free one-month extension. How about a six-month contract?

아이가 잘 먹을지 안 먹을지 모르니까 우선 한 달만 신청할게요.
I will subscribe for one month first because I don’t know if my child will eat well or not.

Related words
밀리리터(ml) milliliter(ml)
-용 for
유기농 organic
희다 to be white

Source: Yonsei University. Korean Language Institute