H-1B status is employer specific and employees are not entitled to perform services for any other employer unless that employer files an H-1B petition. UCI must file an H-1B petition to USCIS on behalf of the employee to allow him/her employment at UCI.
Employee should not complete employment with the current employer (including continue to be paid) until the anticipated start date and until the UCI petition is filed. UCI should be able to exercise the H-1B portability provision of the current AC21 law, which allows the employee to begin employment with UCI based upon confirmation that the H-1B petition was filed and received by USCIS.
For more information, please see https://www.uscis.gov/i-9-central/75-h-1b-specialty-occupations .
To start H-1B Change of Employer case, please see H-1B – Before You Sponsor