Item 113 4fr detail.
Vanity light extension box.
I have though seen light boxes of a row of bulbs that would be used above the vanity and that box sticks out the depth of a surface mounted vanity.
My vanity light is centered above the sink.
Sigma electric weatherproof extension ring 2 gang gray metal weatherproof new work standard box extender exterior electrical box.
E26 e27 light bulb socket extender adapter e26 e27 to e26 e27 flexible extension all directional light adjustable converter for a19 led lamp recessed spotlight etc.
Generally any wall mounted lights will supply enough light even if they are flush mounted and the protruding surface mounted vanity is used if i am reading your description correctly.
It is a modern type bar with 4 lamps attached like this elk lighting inc.
You just have to take the fixture off and see what shape size is the original box.
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