Once the ginger is peeled, continue using the peeler to skim very thin pieces off of the root. Collect all of your pieces is a bowl and add the salt. Massage the ginger for a minute or two to work in the salt. Let it sit for an hour or so.
Once the ginger has sat, bring a pot of water to boil. Pop the beet slice into the pot of water and add the ginger to the boiling water. Let it cook for 3-4 minutes. Once the time is up, drain the ginger and remove the beet slice.
In a glass jar pour the sugar and about a tablespoon of each of the vinegars. Add the ginger and more vinegar until the ginger is just completely covered. Let the ginger sit overnight.