Generally called smeared mortar, this is taking hold in the press recently - was there a Martha Stewart episode on it or something ? This used to be considered a sign of sloppy workmanship, by not striking off or detenting the mortar surface - called tooling or tucking.
Or "German Smear" finish - which term came from sloppy mortar work done by immigrants in Germany and was considered equivalent to "trailer trash" living conditions - now haute arte ! Reminds me of the brick or cinder block industrial buildings converted into lofts in the 70's, where people were paying exhorbitant rates to basically live in an unfinished warehouse or factory !
Anyway - yes, a Masonry contractor can do that. Generally recommended strongly that you not cover the bricks up with an actual "layer" of mortar - as opposed to smearing some over it like a whitewash, to leave a free surface for the bricks to evaporate moisture. Expect to have a fair amount breaking off over time, and some popping out from time to time - which will not hurt the appearance much, but worries some people to see bits and pieces of mortar below their wall or foundation. This will be expecially prevalent if the mortar had been sealed - in which case is much better to sandblast it clean before retucking the mortar joints to a rough squeeze-out or "stone wall" or "unfinished extruded" joint profile.
Going to be pricey, because the normal way to do it is to add mortar and strike the joints off flush, then use a grout bag to place the mortar that sticks out of the joint - basically using an overstrength frosting bag, so very labor intensive.
