We add a lot of the thinly-sliced stems to the chard, when cooking or freezing. One of the things I like about chard is that the cooked vegetable has more substance than cooked spinach, almost meaty - and adding the chopped stem pieces reinforces that.
But we seldom use all the stems, and when the chard is coming on strong late in the season, there is more stem than we can use. DD came up with the idea of using the stems in a cream of celery soup recipe, and we tried it - it wasn't bad. The second time we added some of the leaves to green up the soup, and it was even better. We'll be doing more experimentation with that soup this year, using different fresh vegetables & seasonings.