It's scale, a nasty pest. Read more about them here. I've found horticultural oil to be effective, but it's tricky to use this time of year with the heat. You can use a water spray and gently rub them off the stems, too.

would cutting the affected stem of the plant help?

It might help, but as scale is mobile in its first stage, they'd likely show up other places on the plant. The nuclear option is to use a systemic insecticide, but only if this is an ornamental plant.

Unfortunately this is an edible, herb that we use in cooking :(

I hate scale. The remedies are horticultural oil sprayed on them (to suffocate the little devils), alcohol wiped directly on the areas then rinsed off, systemic insecticide (maybe) or even soap and water. I would try strong soapy water (like dish soap) and really wash the plant thoroughly, then rinse really well and see if that helps. You can tell they're dead if you rub your hand along the stem and they just fall off. Otherwise, they stick like glue. I have to say that it looks like a really bad infestation so you may be fighting a losing cause. Sometimes you just have to start over.