Hello,

I have to first caution you that cutting roots you run the risk of making the tree unstable and in a big wind event the tree is likely to fall over if it has not had enough time to re-anchor itself.  You are also removing the tree's ability to seek out nutrients and water to support its growth.   This job will result in tree stress that may have an impact on the overall health of the tree and make it more susceptible to insect and disease infestation.

Roots should be cut in the fall. Obviously you want to cut as far away from the trunk as possible. 

Make clean cuts by removing soil away from the root only where you want to cut so that a cut can be made without disturbing the rest of the roots.  Leave the roots in place.  Don't lift the root to be cut before cutting. After cutting, allow the cut end of the root to dry out for a few hours to a half day before covering over again with soil. I do not believe that you need to remove the cut root from the soil- It will rot away.   Pulling it up will create unnecessary root disturbance.