는 것은 and 에요
Take the following sentence:

제가 좋아하는 것은 보이차와 녹차에요.

1. Can someone explain why there are two 는 in this sentence, as opposed to just 좋아하는 것?

2. Is this sentence supposed to end in 에요 as opposed to 예요? If so, why?

Question

는 것은 and 에요
Take the following sentence:

제가 좋아하는 것은 보이차와 녹차에요.

1. Can someone explain why there are two 는 in this sentence, as opposed to just 좋아하는 것?

2. Is this sentence supposed to end in 에요 as opposed to 예요? If so, why?

Answer

1. 
There are two major usages of 은/는.

First, 은/는 is used for topic marker.
ex)
이것은 녹차에요.
This is a green tea.

Second, 은/는 used for making an verb or adjective into determiner.
ex)
좋아하는 것을 말해주세요.
Tell me what you like.

제가 그것을 좋아합니다. +  그것은 보이차와 녹차에요.
=> 제가 좋아하는 것은 보이차와 녹차에요.

2. 
예요 is a short form of 이에요.
이에요 is composed of 이다 + 에요

책상이다 + 에요 = 책상이에요
가방이다 + 에요 = 가방이에요
껌이다 + 에요 = 껌이에요
가위이다 + 에요 = 가위이에요
종이다 + 에요 = 종이이에요

If the word before  이에요 doesn't have final consonant(받침), 이에요 can be shorten to 예요.

책상이에요(O) / 책상예요(X) with 받침(ㅇ)
가방이에요(O) / 가방예요(X) with 받침(ㅇ)
껌이에요(O) / 껌예요(X)  with 받침(ㅁ)
가위이에요(O) / 가위예요(O)
종이이에요(O) / 종이예요(O)

So,녹차에요 is not correct.
녹차이에요 or 녹차예요 is correct.