I am afraid you have gotten some inaccurate information about EAD processing being for "any kind of application". There is not a single rule covering every kind of case in immigration, which is part of what makes it quite complex. It very much depends on the kind of application filed as to whether an applicant is eligible for employment authorization while the application is pending or must wait until its approval.
Although I have heard some rumors of a proposal to change the process to grant employment authorization to U visa applicants while their applications are pending, I am not aware of any final changes in the process so far. My U visa clients continue to receive their work permits once their cases are approved, although the work permits do usually arrive a couple weeks before the U visa approval notices. Thus, I would anticipate that your U visa work permit will not issue unless/until your application is approved as well.
Also, keep in mind that processing times do change frequently and are an "average" time rather than the actual time for every case. Even applications that are "accurate and properly filed" may generate requests for additional evidence or require extra time on security clearances. In my experience, the average U visa case filed within the last year seems to be taking between 7 and 10 months for processing, but some take even more time. I hope yours is among the quicker ones.
Ms. Doerrie's answer to your question is general in nature, as not all facts and circumstances relating to the specific person(s) and situations involved are known to her. Ms. Doerrie recommends consulting with an immigration attorney regarding your specific facts and circumstances prior to making any legal decision or submitting any form or application. This response does not create an attorney/client relationship.
Employment authorization is part of the approval of the U visa; it cannot be applied for separately.
This answer is of a general nature and should not be relied upon as final, nor is it intended as legal advice. A consultation with an experienced attorney is always the best way to go.