Yup! It;s called board and batten,board on batten or board on board depending on who you ask. The construction is exactly how it sounds. Wide boards are installed leaving a space between each. When all wide boards are up,narrow boards are nailed on to conceal crarcks. You didn't ask but I wouldn't let either of those fence companies within 5 miles of my house,much less hire them. Consider yourself lucky they didn't do any work for you.

Thank you for your responses! Would you have a idea of the width of the boards are?

No and while it's possible with some effort to work it out,it would be easier if you just took a ruler and measured them.

I suspect that is a sheet product like plywood or cement board not individual boards. I would use 6x6 support posts, four 2x4 horizontal rails, skin both sides with 1/2” cement board vertically, then 1 1/2” battens to hide the screws. This will be an expensive fence but will look great. These fence guys you called are used to standard fences they can build in a day.

You can get board and batten fence panels already pre-made, Trex also makes the composite board and batten sections but the slats are wider. Shop around and try to find a distributor locally or have it shipped, but it will not be cheap, or like HALLETT & Co. mentioned you can build it onsite, which is costlier.

Good luck