What you're doing is sweating the onions: cooking them over low heat in some fat until they become soft and translucent. The thing about sweating is that you either want to do it very moderately, as Pr0lapse suggested, or you want to do it for a very long time to allow them to develop a stronger and deeper flavor. As noted, you don't want to allow them to caramelize, but in this case caramelization doesn't have to do with how cooked the thing is, but rather the level of heat you have on under it.
I would recommend the following procedure:
1. Sweat diced onion and sliced garlic together until soft. Season them with salt early on to aid in the releasing of liquid, and not with sugar at all. Deglaze with white wine and reduce until the alcohol's gone but not the acidity, usually around 2/3rds.
2. Add whatever other veg you think will cook at around the same rate as per your usual procedure, and your stock cube and whatever else. Don't let the onions brown. Make sure to season with salt and pepper, keeping in mind that if you let the liquid reduce much at all it's going to become saltier as it does.
3. Cook the chicken separately from the rest of the soup as suggested. You don't have to cook with butter if you don't want to, though if I lived in Ireland I would take liberal advantage of the fresh dairy, which is seriously among the world's best. Do you not drink milk?
Anyway, searing your chicken will help it develop deeper flavors than simmering it, and it helps you control the degree of doneness. In general, when I make a soup with a lot of individual components I think it's best to cook them separately and put them together at the end, but in this case just separating the meat and veg should do you fine. Just season it on both sides with salt and pepper, lay it in whatever cooking fat you choose that's been heated up in a saute pan (if you don't have a hot pan and hot fat, the chicken will stick to the pan), and flip it about halfway through, when the white, 'cooked' color has creeped up around the sides. After the initial heating of the pan you can let it go on medium. If you don't object to adding butter, add it at this point and use a spoon to baste your chicken with it once it melts; this will also speed up the cooking process. Adding it too soon causes it to brown and introduce bitter flavors. In this particular case, though, I might go with your initial instinct and omit the butter altogether.
4. Chunk up your chicken and add it at the very end of the simmering process, or just place it in the bottom of your bowl and pour your soup on over it.
If you use garlic puree instead of fresh garlic, add it towards the end of the simmering process, as that garlic has already been blanched and simmered to remove its fresh flavor and will not flavor the soup much at all if you put it in at the beginning. Make sure to stir it in very thoroughly.
This is not, by the way, how I would go about making a chicken soup, but it should work to modify your current practices to end up with a better result.