How to stabilize soft nougat?

The stickiness/softness/runniness of nougat is directly controlled by the temperature you heat the water/sugar mixture to. The boiling temperature of a water/sugar mixture depends on how concentrated the mixture is; the more sugar per amount of water, the higher the boiling temperature. As you evaporate the water the mixture will get more concentrated and the boiling temperature will rise. So, the higher the temperature, the less runny the results will be.

Once you've mixed in the egg whites, though, you've made your choice, and can't later decide to make it thicker.

It sounds like you want two different materials: a hard nougat for the outside, and a soft nougat for the inside. Perhaps you could make two batches, and dip blocks of the latter (heated to a lower temperature) into a still-melted bowl of the former (heated to a higher temperature)? It might be tough to get it all to work, though...

In my own practice, I just cook the sugar to a higher temperature in order to make a nougat firm enough to not spread.

However, I have recipes suggesting the use of rice paper, and have eaten store-bought nougat that used rice paper on one or two sides of the nougat to hold it together (and minimize sticky fingers for the people eating it.) I can vouch for rice paper tasting just fine with the rest of the nougat.

I tried to make nougat for the first time. The consistency was way too soft. I left it few hours. I just did not want to waste all those ingredients. I pour the way too soft nougat in a glass bowl and add a little bit of corn flour. I put the glass bowl over a pot with boiling water and stir and stir and stir. It reached a nice chewy consistency, and I spread it on a tray until it was cold. That way I could save the "flop". And the children finished the nougat within a few minutes.

..... A distinct option would be to make little sugar cookie cups to place the nougat chunks in. I wouldn't completely enclose the nougat in dough and bake it, but cookie bowls could make a pleasant, customizable delivery system if other options are dissatisfactory.

Just take a stiff, decorator-approved cookie dough, portion, and with a thumb or measuring spoon, press an indent into each cookie dough ball before baking. As long as the dough is designed not to spread too much, you should get a cookie with a little bowl in it.

You could similarly make meringue bowls, which might actually compliment the egg-white based nougat even better.