The most common and most likely answer to this crossword clue is the 12 letter word INTELLECTUAL
I
N
T
E
L
L
E
C
T
U
A
L
More clues leading to the result INTELLECTUAL
Buddy explains!

The straight part
We know that "Egghead" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case egghead = intellectual.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does INTELLECTUAL mean?
INTELLECTUAL noun- An intelligent, learned person, especially one who discourses about learned matters.
- The intellect or understanding; mental powers or faculties.
- Belonging to, or performed by, the intellect; mental or cognitive.
- Endowed with intellect; having the power of understanding; having capacity for the higher forms of knowledge or thought; characterized by intelligence or mental capacity
- Suitable for exercising the intellect; formed by, and existing for, the intellect alone; perceived by the intellect
- Relating to the understanding; treating of the mind.
- Spiritual.