You'll get cold sweat that way -- then you REALLY freeze.

For a DIY approach not using shoe covers (the preferred approach), cannibalize some socks and put them over your shoes. Obviously, you will need to cut holes for the cleats. Thick hiking socks are best.

Do not try to wear a bunch of pairs of socks in your normal shoes. You'll just cut off circulation to your feet and make them freeze.

For emergencies, carry hand warmers in your pack. If your feet get too cold, you can throw them in your socks for a bit while you defrost. You won't be able to ride with those in there because they're too bulky, but it's a temporary fix.