You should be able to make one out of beadboard if you frame it like a inset paneled door (The opening and the door itself) and use cabinet hinges made for inset doors. I would make the door opening so it lines up with the vertical grooves in the beadboard and the top of the base board so three of the four edges of the door are camouflaged by either a groove or trim. It would probably be neater to make the door out of a separate piece of beadboard (not the one cut out of the opening) so you can start with one slightly larger and carefully fit it for nice clean edges all around.

Then use some kind of edge pull that barely shows for opening and closing it. I have found the hidden push latches get out a alignment and the door sits out of flush, and fingerprints mark the area where you push to open and close.

Here is a hidden door I did, although it is a full-sized linen closet door.

Thank you for the feedback. I do think I'm out of my depth with this, though. I had to look up what an inset panel door is! I found examples of that type of door with a beadboard panel. Do you envision the frame of the door to be like a regular frame, with the panel being beadboard? The frame will make it more visible but that may be what we need to do. The beadboard is only 1/4" thick so not very structurally robust to be used on its own. My countertop installers are not likely going to be up for creative carpentry to solve this problem for me on Wed, so what I may need to do is get the shelf put in the recess in the support wall, let them go ahead and install the raised bar, finish out the wall (cutting me a hole to access the shelf) and then I can get with a carpenter to finish out what I want on the opening and door.

No, the frame will be on the back of the beadboard panel to make it sturdy. The idea is to make the door a size that the sides align with grooves in the beadboard, and the bottom aligns with the top of baseboard. Only the top will show as a horizontal line across the beadboard. It will take a certain amount of finesse to have it be as invisible as you envision.

Okay, I see what you mean now and that makes sense. Would the reinforced beadboard door sit flush in the cutout in the wall, or would it be slightly larger than the cutout and therefore rest on the outside when closed?

The question you should ask yourself is what are you going to store in there? Is it going to be worth the extra money and effort to have the small storage area?

Personally I do not think I would like the look of it because even if you get the beadboard panel flush, you will still be able to see the horizontal line of the opening I believe. We have a peninsula layout and plan to put access to a blind cabinet on that side, but we added panels along the peninsula to match the door so it looks uniform. If we could not have done that we probably would have just closed up that space and made it inaccessible.

All flush.

I would plan to store glass pitchers, large glasses we use infrequently, etc. Stuff I would like to have handy in the kitchen but don't need in my primary cabinet space. I've already bought the shelf so I'm at least somewhat committed. If we do as palimpsest suggests and align 3 of the 4 sides with "lines" that will already exist, and we keep the door flush, I'm not too worried about it looking bad. This is down at the far end of the peninsula - next to the wall - and there will be a barstool in front of it. I don't think it will be noticeable and I do think I would put the shelf to good use.

I had a similar "found space" on the back side on our angled peninsula. There is a triangluar space between the straight run of cabinets, and the angled peninsula. There is a large panel across the back of the entire peninsula. To make the space accessible, I mounted the panel to the back of the peninsula, using side-mount drawer slides. Now you can slide the panel to the side, and access the space, or slide it back, and have no idea that it is there.

I love squeezing out that last bit of storage space. I'm a practical person who gets more jollies from function rather than form, so here we are! I had to post a pic of the shelving unit I just picked up from the woodshop. I just slipped it into the recess in the wall, it's not officially installed yet. The two shelves are adjustable. I was also looking at where that one visible edge will be, and I think the overhang on the bar will really help mask it. When you are standing I'll bet you won't be able to see it. If you are sitting across the room you will probably be able to pick it out, though, if you look. I'm just going to trust that a good contractor will be able to fashion something that will work well as a hidden door. I'm not looking to make it invisible, just unobtrusive.