How to make savoury brioche bread

Brioche is not necessarily sweet. By definition it is a bread with a high butter and egg content...an enriched dough. Also, salt does not necessarily = savory. Umami defines savoriness. Fermentation increases umami. It is one of the reasons we enjoy bread in the first place. Adding more salt will just make your recipe more salty. I would consider reducing the sugar, and consider the quality of the milk, butter, and flour you are using.

I can think of a couple of ways of increasing the savoury flavour of brioche.

First, reduce the sugar. As Moscafj noted, Brioche is a rich bread, and made so by the presence of lots of fats/oils and protein from the eggs and butter. It does not need to be sweet necessarily. I think you could easily halve or quarter the sugar without affecting the end result too much, it certainly won't affect the rise.

Second - add some savoury flavouring. You may be able to do this by using a cultured butter, which are less sweet than regular butters, but I doubt that your favourite burger place is doing this. What I would think they are doing is using a less sweet dough and adding a savoury compound, probably as a wash to the bun before baking, but maybe added into the dough.

For the savoury flavour, I would start with onion powder. This should add a slightly sweet, but savoury flavour to whatever it is added to. If this is on the outside of the bread, it will be the first thing encountering your tongue/palate when you bite into the bun. I would mix onion powder into an egg wash and brush over the top of the buns. This will make the bun glossy and have that umami hit you are looking for. You could also try garlic powder, or a mix of the two, but be careful with either so as to not make them over-powering.

Lastly, you could add a savoury seed to the outside; sesame, poppy seeds or even chia could work.

From Wikipedia:

Brioche is considered a Viennoiserie because it is made in the same basic way as bread, but has the richer aspect of a pastry because of the extra addition of eggs, butter, liquid (milk, water, cream, and, sometimes, brandy) and occasionally sugar.

So basically, the additional sugar is not necessary for your bread to be (and taste like) brioche; even many recipes online includes the sugar to be optional.

Although this is not traditional, I find that adding a bit onion powder into bread makes them smell and taste really savory. I would recommend you trying that, or adding just a little bit herbs to the dough.

_{(When using a bread machine I would sometimes add large chunks of fresh onion, as the paddle in the machine grinds it down well when it mixes & kneads the dough).}

I've been thinking about this some more from the point of view that you're trying to imitate something commercial.

One thing with commercial cooking is that you can never rule out outright cheating. It wouldn't surprise me if the baker was adding monosodium glutamateto the dough. This is an artificial source of umami (savoury) flavours. They'd still need to reduce the sugar, as MSG+sweetness has a slightly odd flavour

You can buy it as a food additive. It's not something I cook with, so at home I'd try a tiny bit of yeast extract instead - not enough that you'd really taste it, at least at first. I think this would be a more suitable source than soy sauce.