Yep, my hunch was right. I just googled this:
CCE - Suffolk County: Spinach Leafminer
"The spinach leafminer attacks spinach, beet, sugar beet, Swiss chard and many weeds including lamb's-quarters, chickweed and nightshade.
The maggots feed between the upper and lower leaf surfaces of the host plants mining out the tissue in between.
Because this insect overwinters as a pupa in the soil near where the crop was infested the previous year,
crop rotation should be practiced. Also cover plants with fine netting/cheesecloth/floating fleece to protect them from adult egg-laying flies.
Remove infested leaves before the larvae drop to the soil. Do not compost the leaves:larvae might continue to develop and emerge as adults to reinfest crops.
Control weeds. Destroying the insects' wild food plants should also be helpful in reducing the numbers of leafminers. This includes lamb's-quarters, chickweed and nightshade in and around the garden."