Can employee leave within 2 months of job and UAE...

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As soon as your residence visa expires in UAE, the sponsor has to make it quick to the renewal of residence his / her sponsored / employees within the time-limit for renewing visas, so that to avoid any imposed fines or legal consequences.
According to the law, the general directorate of residence and foreign affairs (GDRFA) grants a 30-day as a grace period to stay legally in the UAE after the expiry of the residence visa.
According to the fines system in the ministry of the interior in the UAE, the fine for overstaying in the country after the expiry of the grace period:
25 AED per day during the first six months (180 days)
50 AED per day during the next six months
100 AED per day after more than a year.

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We would like to inform that Indeed, your behavior is correct in making an escape notice against the employee, since she is considered a fugitive if he stops working for more than a week for no legitimate reason and without the employer's knowledge of his whereabouts.
Procedures of his communication
The employer or his representative:
fill out the escape notice form and submit it to the Department of Labor to which the facility belongs
Prove that the establishment has no previous fines, or that it paid all fines to provide a bank guarantee worth three thousand (3000) dirhams for each worker is to report his escape and the escape letter contains the following:
- Cancellation of the worker's guarantee
- Deprivation of a worker from the UAE for one year
- Compelling the establishment to pay the worker's wages and cancellation fees
- Transfer of establishment to category (c)
- Compelling the establishment to pay ten thousand (10000) dirhams with the fines due. 
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