The work Machshireh Mila (by Hacham Eliyahu Shama Ha’levi, Chief Rabbi of Aleppo, d. 1814) mentions the custom to pronounce at a Berit Mila the Beracha, "Yesimcha Elokim Ke’Efrayim Ve’chi’Menasheh" – "May G-d make you like Efrayim and like Menasheh." Following the circumcision, after the recitation of the blessing in which the baby is named, the father places his hands on the child’s head and pronounces this blessing.
This custom, as the Machshireh Mila notes, originates from the ancient Targum Yonatan Ben Uziel translation of the Humash. The Torah tells in Parashat Vayehi (Bereshit 48:20) that before Yaakob Abinu’s death, he proclaimed that forever more, parents would bless their children with this blessing – "Yesimcha Elokim Ke’Efrayim Ve’chi’Menasheh." Targum Yonatan (listen to audio recording for precise citation) explains this to mean that this blessing would be pronounced at a child’s Berit. This verse thus forms the basis of the practice to confer upon an infant the blessing of "Yesimcha Elokim Ke’Efrayim Ve’chi’Menasheh" at his Berit.
A beautiful explanation for this custom is offered by the Minhat Elazar (Rav Haim Elazar Spira of Munkatch, 1871-1937). The Talmud says that the human soul would have been better off remaining in the heavens, where it would stay pure and pristine, rather than coming down to this world, where it is likely to make many mistakes. However, once a soul has descended into this world, it should strive to make a Kiddush Hashem – to bring honor and glory to Hashem here in our world. This means that the truly significant moment for a person is not the moment of birth, but the moments when he creates a Kiddush Hashem. Thus, age is far less important than accomplishments; what matters most is not when a person entered the world, but the extent to which he brings glory to Hashem during his stay in this world. This is expressed by the fact that in the blessing of "Yesimcha Elokim," Yaakob mentioned Efrayim – the younger brother – before the older brother, Menasheh. The reason, as the Sages explain, is because Efrayim produced Yehoshua, who created the greatest Kiddush Hashem possible when he had the sun stop in the sky, thereby demonstrating Hashem’s power to the entire world. Mentioning Efrayim’s name in this verse before Menasheh’s name demonstrates that what matters is not when a person was born, but rather the Kiddush Hashem that he makes during his lifetime.
Therefore, specifically at a Berit Mila, and not at the moment of birth, the father gives his child this Beracha. After the Berit, when the infant has, for the first time in his life, achieved a Kiddush Hashem (albeit passively), he is told that this is what matters most – his accomplishments during his lifetime. This Beracha is given at a Berit to convey this lesson, that the most significant moment is not birth, but rather the moments in which a person brings glory to the Creator.
Summary: It is customary at a Berit that after the recitation of the Beracha in which the child is named, the father places his hands on the child’s head and pronounces the Beracha, "Yesimcha Elokim Ke’Efrayim Ve’chi’Menasheh."