Just saw this on the Internet. I don't know if this is sound advice:
"This looks like leaf miner damage. Discard affected leaves preferably in plastic bags because the larva are inside chewing paths. Watch for little white eggs on underside of leaves. I remove them by dipping a cloth in soapy water and wiping the leaves. I also spray with soapy water in case any adults are around. You have to inspect often and stay on top of it."
"Leaf miners. Cut the leaves off, throw them out, and your beets will grow new leaves"