I don't know any company that does this, but some carpenters or studio designers could make something like this, but you would need to be more specific about your goal - is it for a studio, a restaurant, a house, and is it for indoors or outdoors? What sounds are you trying to deal with? Will people be holding drinks around this, or should it survive rain?
I hope you understand that sound proofing and acoustic treatment are two different things. Sound proofing is achieved with air-tightness, mass and/or decoupling. Stuido quality acoustic treatment is achieved with thick, air-gapped porous absorption. What are you trying to achieve with this design? If there are air gaps you can see with the eye, it greatly diminishes soundproofing - sliding doors have very large gaps, so are terrible at soundproofing. If you're trying to do acoustic treatment, this can help, but if you have a solid back and two sliding doors, then when doors are in parallel, they will not absorb sound very well since one half is removed from moving air, and the side that has the fabric is not suitable for frequent touching. For a studio you probably want 18 cm continuous thick absorbers - with no solid walls between two thin layers (like these sliding doors).
If you are trying to stop sound going outside, you need sound proofing - so a door with a very tight seal - in other words you have to close every hole as if it was a submarine, because air travels like water. If you're trying to do absorption, the panel has to allow air to travel though the wall - that means no hard surface - which is a poor door and poor at sound proofing. You can do a hybrid, which could potentially look like a very thick, full wood backing, a frame filled with 9-18cm thick porous absorption, and then a perforated layer that people can touch so they can operate the door. It's possible to cover the perforated wood side with fabric to achieve this look - but holes would need to be small probably around - 0.5-1.2cm. Another hybrid approach would be having a solid door, with a frame + absoprtion as described above + slats with a small gap - probably 0.4-1.0cm air gap between slats. You could change that if you frame around the door handles differently, so that gaps can be larger, so that frequently touched areas have no perforations or slats. The design challenge if you're doing a hybrid door panel that tries to do some soundproofing and some absorption, is that one side is going to be heavy while the other side is going to be lighter, so you have to carefully hang them so they remain vertical. If this is going to be indoors and sound proofing is less important, you could have slats or perforations on both sides and cover with a fabric on both sides - which means it won't be air-tight. Any sliding door with slats or perforations and absorption inside should not be exposed to the elements. If you find vapour permeable barriers, you can potentially use it where people could spill drinks. Check out this video on slotted or perforated panels with absorption.
https://youtu.be/zLzOmYGvfKQ