I only mow chicory if I need to keep some weed from going to seed or I might occasionally mow it high in late summer to get some tender new fall growth going.
I’ve seen deer take the top 18 inches out of chicory plants that are nearly five feet high, so it remains palatable for a long time after it gets tall. This is assuming that you are growing a good variety of forage chicory.
If you mow it too short or too often you could be mowing down food that deer will readily eat, and any time you mow a tall broadleaf plant like chicory, you are giving the advantage to grasses that are growing in your plot.
The two varieties I feel the best about from personal experience are Commander and WI, but there are others that will probably work just as well.
I might also add that chicory is a nitrogen lover, so the perfect companion plant is clover. I don’t think you can beat a good plot of chicory mixed with medium red and ladino clover in my area for year round attraction except maybe late winter. Even then I have seen them pull chicory plants out of the ground and eat the roots.