Learn English – Too deep or too deeply

They mean two different things; which one you want depends on which meaning you want.
_{As a side note, it should be written as The Atlantic Ocean rather than just Atlantic Ocean}

The Atlantic Ocean has a deeply indented coast line which facilitates trading and other mercantile activities.

This indicates a greater magnitude of indentation of the coast line.

The Atlantic Ocean has a deep indented coast line which facilitates trading and other mercantile activities.

This would likely be written with a comma (deep, indeted coast line), and means that the coast line is both deep and indented.

In principle, deep is an adjective (modifies nouns), and deeply is an adverb (modifies verbs). In practice,

He cut deep into something. (about 2190 results in Google Books)
He cut deeply into it. (1030)

By which you can see that native speakers tend to ignore that principle when the verb is being used "actively". But when it's just a "passive" past participle, we invariably follow the rules...

He was deep hurt (1)
He was deep wounded (4)
He was deeply hurt (55000)
He was deeply wounded (20400)

Also note that in some constructions that might superficially look like "adverbial" usages, the word is actually part of an adjectival "modifier of location". We use the simpler form there as well...

He walked deeply into the forest. (38)
He walked deep into the forest. (7390)

In OP's example, deep is used adjectivally as per the above (it's telling us where the chicken was buried). It would be perfectly grammatical - if somewhat less common - to use deeply there (in which case the emphasis would be on how she buried it, not where).

This use of superficially adjectival forms in adverbial contexts is quite common in English. I can't think of an example where the difference between deep/deeply would be perceived as changing the meaning, but this certainly occurs with other words. For example...

1:He acts bad.
2:He acts badly.

...where in the absence of any other contextual information, we naturally tend to assume that #1 means he acts like a bad person, whereas #2 is likely to be interpreted as he's not good at acting.