Last year, some critter was eating carrots and other root crops in the ground. When pulled, half of the root was gone. These veggies are in raised beds in southern NH, zone 5a. Suggestions on what critter & how to prevent such losses this year?

Do you see tunnel paths in the grass? Where are they heading? Plop a few onion sets in the planted rows, I've found that deters them little mole guys. Even if onions aren't companions to some plants, it's better than having no food.

Put the onions as far as your finger can poke them.

No mole tunnels in sight. I thought it might be moles, but reseached moles & found that moles primarily(exclusively?) eat grubs & other critters.

The critters are voles...they love beets, carrots and a lot of other plants and bulbs.
Try planting onions around your carrot bed!
I've noticed a big reduction in vole damage when I plant alliums near their favorite foods.

I tend to get grubs that eat my root crops....but they leave little 'roads' all over the root.

.....you need a barrier of garlic planted around the perimeter....