A new generation of startups is racing to solve a puzzle: how to make a juicy burger without harming a single steer.

Food scientists are mixing pea, soybean and beet molecules into concoctions that mimic ground beef. Bioengineers are growing muscle tissue from animal cells in steel tanks. The question is whether either method can find favor beyond vegetarians and animal-rights activists among the U.S. consumers who overwhelmingly identify themselves as carnivores.

“The...