The Ministry of Labour (MOL) or Labour Department in Dubai and the UAE governs all work related issues. The MOL is responsible for issuing work permits (or labour cards) and imposing labour bans on those who are entitled to one. The Ministry of Labor does not issue residence visas or entry permits, they are dealt with by the relevant UAE immigration department (in Dubai, the DNRD).
An exception to this is labour permits for maids, they are processed at the immigration department along with their residency permit.
Note that an Employment Entry Permit (often called an employment visa) is not a labour card or work permit, it is an entry permit issued by the immigration department for entry to the UAE temporarily (30 days) and allows the holder to work for the designated employer while the employer arranges processing of the labour card and residence permit (residence visa). However, the immigration department does not issue the entry permit until the MOL has approved (not issued) a work permit for the employee. That's why it takes a few weeks to get things organised.