The most common and most likely answer to this crossword clue is the 4 letter word FAST
F
A
S
T
More clues leading to the result FAST
Buddy explains!

The straight part
We know that "Don't eat/quickly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case don't eat/quickly = fast.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 FAST mean?
FAST noun- A train that calls at only some stations it passes between its origin and destination, typically just the principal stations
- Firmly or securely fixed in place; stable.
- Firm against attack; fortified by nature or art; impregnable; strong.
- (of people) Steadfast, with unwavering feeling. (Now mostly in set phrases like fast friend(s).)
- Moving with great speed, or capable of doing so; swift, rapid.
- Causing unusual rapidity of play or action.
- (of a piece of hardware) Able to transfer data in a short period of time.
- Deep or sound (of sleep); fast asleep (of people).
- (of dyes or colours) Not running or fading when subjected to detrimental conditions such as wetness or intense light; permanent.
- Tenacious; retentive.
- Having an extravagant lifestyle or immoral habits.
- Ahead of the correct time or schedule.
- (of photographic film) More sensitive to light than average.
- In a firm or secure manner, securely; in such a way as not to be moved; safe, sound .
- (of sleeping) Deeply or soundly .
- Immediately following in place or time; close, very near .
- Quickly, with great speed; within a short time .
- Ahead of the correct time or schedule.
- Short for "stand fast", a warning not to pass between the arrow and the target
- The act or practice of abstaining from food or of eating very little food.
- The period of time during which one abstains from or eats very little food.
- To restrict one’s personal consumption, generally of food, but sometimes other things, in various manners (totally, temporally, by avoiding particular items), often for religious or medical reasons.