An industry-wide oversupply of chicken breasts, wings and thighs is squeezing poultry processors' profits.

Americans are buying more chicken at the grocery store, and restaurants are increasingly adding more chicken-which is cheaper than beef or pork-to their menus to cut costs, Pilgrim's Pride CEO Fabio Sandri said on a call with investors today. But that demand wasn't enough to offset a 4.5% increase in supply during the company's second quarter, he said.

Wholesale boneless skinless chicken breast prices are down about 34% from 12 months ago, according to FactSet. That's putting pressure on companies like Pilgrim's, the second-largest poultry processor in the U.S., which posted lower profit in its latest quarter.

Chicken companies ramped up production over the past year in anticipation of rising demand as beef prices soar. But they appear to have overshot as more birds than expected reached maturity in recent months, and after companies found a way to solve an issue that limited the number of chicks they could hatch.

"The headcount was a little higher than everybody anticipated," Sandri said.