Do you let an infant “cry it out”?

Today, my 10 week old just wouldn’t stop crying. He cried off and on for nearly 4 hours, even after being fed and changed. My fiancé told me to let him cry it out. Well, letting him cry it out turned into him gasping for air because he cried so hard. It took him a good ten minutes to catch his breath after I picked him up. I feel shattered. I’m never letting that happen again. He finally fell asleep, but I didn’t want to let him out of my arms. I’m seriously so upset about how “crying it out” went. Just curious what you ladies do