Interresting idea, I'd love to know if you can grow mushrooms from the supermarket mushrooms.
I guess you can try it yourself and see if it grows or not, I don't know how to grow mushrooms.
We used to have some mushrooms in our garden in the shady areas under other plants where it's wet, even in mid summer when the tempeture is around 35ðC - 42ðC
I didn't see any of it grow in years though.

You should be able to collect spores from the most mature chanterelles by placing them on gills side down for a few hours. After collecting the spores, you can still eat the flesh of the mushroom. The spores drop and make visible colored gill patterns on paper. I use spore prints as an additional identification technique for mushrooms that I forage in my area. I used to display some of the more beautiful spore prints as artwork. You can plant the spores in your growth media. I believe that chanterelles are more difficult to culture compared to other mushroom types.

If the loose mushrooms at farmer's markets or Whole Foods type of grocers have what looks like 'roots' on them, they can be planted or use for mycelium. YouTube has videos about it - I watched with glee.

Keep us posted on how this works out for you. I am hoping to spread my chanterelles to a few more areas in my yard doing something similar. I already started rubbing the old, inedible, yucky oyster mushrooms on some almost dead or already dead trees to propagate them. Too early to know the results, though.

The only experience I've had with this has been with the 'inky' mushrooms. We lucked out and got some great organic compost from Lowes of all places, it was in a black and pink bag, and in the later summer I noticed that I had inkies coming up in the pots! I sautéed them in butter and sea salt, very good! I then took the expended soil after the season was over and planted it in the garden, so I hope they spread more. Not sure if the chanterelles would work, but it is worth trying. I know the instructions for oyster mushrooms say that after the little mushroom kit you grow is 'spent', you can cut logs horizontally (make sure they are from living trees, let the logs sit for a month, and then put the spent spawn between the logs in a shaded place and water. Supposedly it will produce more mushrooms. I think it's best to do this just before a rainy season, and cut the trees if possible when they are dormant for more nutrients. I think trees that oysters love are maples.