Caramelising is a chemical process in which sugars decompose under the influence of heat (pyrolisis). It happens to any heated sugars, no matter if they are free (as in heating refined sugar for making candy) or bound in something else (such as the sugars naturally occurring in an onion). The outcome of the process are compounds which have a dark color and pleasant aroma.
Sauteeing is a cooking technique. It consists of frying small pieces of solid food on very high temperature with very little fat while shaking the pan all the time, so they won't overheat and/or stick.
Many people don't know the technical meaning of sauteeing and use the word for plain shallow frying at medium temperature and without shaking the pan. This seems to be the definition your employer was using.
In fact, you were shallow frying both kinds of vegetables, which resulted in caramelisation plus other changes for the onions and in these other changes only for the vegetables which don't contain significant amounts of sugar.
Cooking recipes frequently avoid saying just "fry the onions", because onions have to be brought to a different state for different recipes. So they usually use a word which implies a desired final state, such as "caramelize". For many other vegetables, which only have a single usable state of doneness, they specify the technique instead. This is why different words can be used for the same process - one describes the technique you are using, the other describes the changes which are happening, it is like saying 'I am going sunbathing' or 'I am going to catch some tan', which are indeed the same process.
Incidentally, true sauteeing is a bad idea if you want to caramelise onions. Low and slow is the way to go if you want caramelised onions, while during the high temperatures used in sauteeing they go from translucent to burnt without passing a nice caramelized stage. But just leaving them in the pan for some time is a good technique.
You also say grilling. This is a completely different technique, and it is done on a grill instead of in a pan. I can't imagine how you would caramelize cut onions on a grill, unless you put a griddle on the grill, which is equivalent to frying.
My advice: ditch the soaking liquid.
Here's what I just tried. I divided my dried mushrooms up into 10 bowls: 5 with dried chanterelles and 5 with dried porcini. I added equal amounts of water to each at the following temperatures 10°C (directly from the tap), 40°C, 60°C, 80°C and 100°C (or as close as I could get).
After soaking for 15 minutes I sampled the soaking liquid from each bowl.
All samples tasted astringent and bitter. Naturally the taste was less pronounced for the colder liquids, but not so much as I had expected. The liquid that the porcini soaked in tasted somewhat better than that of the chanterelles. None of the soaking liquid tasted good enough that I would add it to a stock; I was almost gagging when the experiment was complete.
I wrung as much liquid as I could out of the rehydrated mushrooms and used them to make a vegetable stock. With the flavor of the soaking liquid fresh in my mind I could still detect a note of bitterness. This was easily dealt with using a little salt.
The bitterness from the soaking liquid, I know from past experience, cannot be completely removed with salt.
Update: Looking at messages from an online forum here that deals with the same problem, I thought I'd look at some suggestions made there.
Mushroom dust is the cause of bitterness.
There was an amount of dust in the dried mushrooms I purchased. However, I rinsed the mushrooms before starting the experiment described above, so dust doesn't account for the bitterness.
Bitterness is due to bruised fruit that has oxidized before it dried.
I'm not 100% sure about this, but wouldn't bruising change the color of the mushroom? I didn't notice any discoloration. Anyhow this is a quality issue and I've more to say about quality later.
Water is the problem; use a water filter.
The water in Stockholm is neither chlorinated nor laden with minerals (I have had the same kettle for years and there's not a hint of mineral deposit inside). This is not the problem.
Mushroom quality is the issue. Use mushrooms that have been dried and stored properly.
Quality seemed like it could be an issue to me. My impression is that supermarket dried mushrooms in Stockholm don't have a high turnover and could have been on the shelves a good while. The mushrooms I used in the tests above had quite a lot of dust in their packages and that surely isn't a good sign.
Today I looked around for something of visibly better quality and found cloud ear and shiitake mushrooms in the local Asian store. These mushrooms look pretty good and their packages contained almost no dust. The dried shiitake were vacuum-sealed in a package together with an oxygen scavenger.
I soaked these mushrooms at 10°C and 80°C for 15 minutes. Cloud ear mushrooms don't have a strong smell and the soaking liquid didn't taste of much after soaking. The little taste I could detect was slightly astringent and not particularly pleasant. Incidentally, cloud ears don't rehydrate well in cold water and were still quite solid after soaking.
Dried shiitake, in contrast, have a strong musty smell and the soaking liquid was relatively strong tasting. I did taste some pleasant mushroomy flavors and for a time I had to weigh whether I liked the taste or not. Unfortunately these flavors were mixed with some definitely off tastes, among them the familiar bitterness. In the end I couldn't stomach more than three tablespoons of the liquid.
I had a notion that mushrooms soaked in cooler water might retain their mushroom flavor whilst leaching their bitter taste. To this end I drained the mushrooms that were soaked cold and added boiling water to them. After a further 15 minutes I taste tested again. Comparing this second soaking liquid to that of the shittake initially soaked at 80°C, the flavor was much more dilute. I'm sad to say that mushroom flavor is lost even when soaking cold.
My advice for now remains the same: discard the soaking liquid. If that strikes you as wasteful I would at least urge you to taste a tablespoonful of the stuff before you decide to add it to your broth.
Best Answer
The only reason I can think of against it is that your dried morels will have dirt in them and will contaminate your broth.
After the mushrooms have been re-hydrated, pick them up careful from the broth (they usually float).
Then, strain the broth using a coffee filter or a fine mesh strainer (@moscafj).
Use the remaining broth for your risotto.