I don't know where "1 unit" comes from, but the Portion menu lists 1 unit as 52g, and 0.5 breast as 86g. So 1 unit is about one third of one. Using the numbers on that site, you'd need 2-2.7 average skinless breasts to reach the target 108-146g, which is a lot more reasonable. And that assumes there's zero other protein being consumed.
I hope your numbers were more accurate when you determined you were struggling with hitting targets....