The leaf miner in my garden loves the beets and chard the best. No wonder, they are sister plants. Leaf miners also bother (to a smaller extent) my spinach and lettuces, but boy, if there's a beet patch around for them to play in, that's where they're going.
I don't exactly know which species of critter I've got here at Love Apple Farm which is the exact "leaf miner," but I do know how to get a handle on them.
Our first line of defense is to trap them with a blue sticky trap, as you see in the pic below. Position it at plant level in the garden, and your offending leaf miner is attracted to the trap instead of the plant, gets stuck on it, and that's the end of it.
We also can hand pick each leaf off that has the miner inside. Be careful what you do with these Don't simply throw them away, the maggots will hatch in the trash (or compost pile) and continue their life cycle. Either put them in a sealed plastic bag or squish each one, killing the maggot. Better yet, give them to your chickens. They'll love eating the leaves AND the maggots. The ultimate lean protein and leafy green diet.
Don't have chickens and wish you did? Consider taking my class in April on Keeping Backyard Chickens. In the meantime, watch out for those leaf miners, they can do some serious damage!