Our chickweed comes out in the very early spring. We eat it in salads, and I also put it in the stockpot when I'm making chicken broth.
We don't care for the stems, so we pull the leaves off. Tip with that- it is soooo much easier to pull the leaves off *before* you wash them! They stick when they are wet.
We also feed extras to the chickens, who love it. I purposely let it go to seed, and shake it all around certain patches for the next year's crop, then throw those greens to the chickens.
The younger ones (before the seed) are yummier.