If you find one of these among your ancestor's papers, congratulations!--you are ready to go on to the Lithuania part of the search! This one belonged to my maternal grandfather. This birth certificate has all the information that I was looking for, and even more!
My Grandfather's Birth Certificate from Pakapė RCC (Found Among His Papers).
This birth certificate was in the possession of my parents, who had many of the papers left by my maternal grandfather.
My father explained to me that after the Social Security Act was passed in 1935, many Lithuanian immigrants sent back to Lithuania for birth certificates because the government said that the birth date on the Social Security account application was subject to verification and there were penalties for submitting the wrong date. He said that, for most Lithuanian immigrants, that was the first time in their lives that it was important to give a correct birth date.
My paternal grandfather might once have had one of these, but his papers had been lost.
Women were less likely to be in the workforce during these initial years of Social Security, and therefore less likely to have sent back to Lithuania for such a birth certificate.
My paternal grandmother died in 1938, and had no such papers.
There also was no such certificate for my maternal grandmother. [Years later, after I had located my grandmother's actual birth record and found out that it had an erroneous maiden name of her mother (an entry error by the priest), my mother revealed to me that my grandmother had once received her birth certificate from Lithuania, but had torn it up into unreadable pieces after she read it. And now we knew why--it had the wrong maiden name of her mother!]