Picture? Are your walls an orange peel texture now? Sand smooth or thin MDF.

(Nail gun with finishing nails to do this job. It will save having to fill nails)

What? There is no backing board for board and batten, only boards and battens.

Are you talking about wainscoting? Board and batten is an exterior siding method.

Many people use 1/4" MDF for the field under the trim for wainscoting. If your trim design allows, nailing the MDF to wall studs allows for easier removal later. Using adhesive(construction adhesive/mastic) will mean removing the sheetrock if the MDF is removed.

Board and batten is what that style of wainscoting is called, and real historic board and batten has a panel of wood (the board) under the battens. These days it is often done on the cheap with battens directly on the wall.

OP, there are instructions on the internet for how to do this - as a previous poster did say, 1/4 inch mdf will do if you plan to paint.