How does one glue together fried onions with an edible, savory adhesive?

It might be worth experimenting with transglutaminase RM, which is an enzyme that glues proteins together. Given the low protein content of onion, I suggest the RM formula, as it has added sodium caseinate. However, this might not be enough additional protein, and you could use extra sodium caseinate, as these folks do for their veggie burgers. I'm not clear on your process, but you might try mixing these ingredients with your onion, refrigerating to set, then fry.

Oooh! This is interesting! I think the first thing I'd like to know is there a requirement for the entire bar to savory? Because a lot of foods mix flavors and textures. Salted Caramel with nuts is an easy example. I can't remember the episode but there was an interview with David Chang of Momofuku on the Splendid Table where he spoke about his grand unified theory of flavor where the most satisfying dishes contain sweet, bitter, salty & sour (often now with hot and umami added in). If you are willing to add in others you expand the range quite a bit as most sticky things are sweet. And you could have just that binding layer be sweet.

If not the next question I'd have centers around what sort of textures are acceptable along with how long until consumption? If, it short term maybe something Tamagoyaki would work. That's more adhesion by pressing into a softer material and obviously wouldn't work if you are thinking bar as in a packaged bar that gets sold in a store.

Modified corn starch and rice are options. You could enrobe or possibly even spray those.

Dough.

I make oatmeal cookies with so many other things in them that the dough can barely hold them together. Chips, nuts, dry fruit, coconut; all that good stuff. I can imagine that strategy here.

You could do the same with your crispy onion. Make dough with butter, flour, egg and baking soda. Mix in the onions (gently! you don't want to smash them!). Or make cookie then gently press in onions to keep them whole. You want so much onion that it is barely a cookie.

You could deep fry these and then you would have a fritter. Or you could bake them into a cookie or bar.

If you add some sugar to the dough it will brown when you bake it. The thing will be converging more on a cookie if it is sweet but the browning is a good savory flavor.

You could add spices appropriate for what you are shooting for. An Indian style fried onion cookie could have curry type spices (cumin, black pepper, cardamom, coriander, ginger, slivered toasted almond etc) and it would be good and highlight the onion star of the show. I would add spice to dough and shake red pepper flakes on top.

In fact I have a bag of fried onions intended to top salads and Mac&cheese. I think I will try this cookie. Will followup.

Instead of starting with the fried onions, you might want to consider frying the onions yourself.

There's a dish called an "onion loaf". (I think I first had it about 30 years ago at Buddy's Crabs & Ribs in Annapolis, MD ... it's no longer on their menu, and I have no idea how wide-spread it is)

As best I can tell, it's a bunch of onion rings (battered & fried rings of onions), that's pressed into a loaf pan when it's still warm, then inverted onto a plate to serve. I managed to find three recipes online that all call for baking for 10+ minutes once they're in the loaf pan:

• https://www.recipetips.com/recipe-cards/t--153116/onion-loaf.asp
• https://www.food.com/recipe/easy-onion-ring-loaf-76405
• https://cookpad.com/uk/recipes/12360557-miller-carter-fake-away-onion-loaf#

If I were going to try to make it more like what you're asking for, I'd probably look to Indian recipes (onion bhaji or onion pakura) where the onions are sliced thinly, fry them up in batches and then salt & squish each batch individually. (maybe a plate, paper towel, fried onions, paper towel, then a weight of some sort)

If you find that you need to bake them further, perhaps fill a sheet pan, then place a second sheet pan on top to squish them down, and then bake it for 5-10 minutes. (it won't need as long as a loaf, as the food is thinner)

And as fried foods don't stay crispy forever, you might want to add something else into the batter to add crunch ... like coarse ground cornmeal.