You are correct that the syllable block 이 that follows 닭 changes its pronunciation. The double final consonant in 닭 is pronounced as a ㄱ-받침. However when this double final consonant is followed by a vowel sound (technically the silent initial consonant ㅇ) the second final consonant moves over and resyllablizes. The first final consonant stays and gets pronounced as its own 받침 (ㄹ in this case).
Hence 닭 is normally pronounced /닥/ in a vacuum, but 닭이 is pronounced /달기/.
Edit: Note it takes a practiced ear to hear the difference between 딸기 ("strawberry") and 닭이 ("chicken", tagged with a subject marker).
The word for "strawberry" is spelled with a tensed consonant, while the word for "chicken" is spelled with a plain/neutral consonants. Its hard to explain the difference text, here's a good video about it. Note that the video also covers the third plosive variant: aspirated.
It may be better to cut of romanisation because I think it's better to "connect" just sounds with the letters in your head instead of other letters they may become if you have something like "으".
So what I wanted to say is that if you read the letters without the imaged english letters, gae and gwae is actually not a question. But I would prefer "gwae" I guess with a double g because it's ㄲ instead of ㄱ.
Also I like your profile picture.
Are we talking about all expensive chicken being cute?
Or of two chicken choices, are we saying the expensive one is cute and not the less expensive chicken?
And are we talking about a live chicken, or a cooked one that the chef decorated to look delicious and cute?
Not talking about all, just one? Then "the" is correct.