It is implied that SEELE conducted an experiment between a human and Adam, and it birthed Kaworu Nagisa, which played a role in the initiation of Second Impact. Is there any reason SEELE decided to perform a contact experiment on Adam?
SEELE didn't perform it, but merely funded Misato's father's research and expedition. In particular, the research on what is latter know as the S2 engine, which was needed in their plan for instrumentality.
Additionally, the political changes that came about from the aftermath of the Second Impact gave the United Nations expanded powers. Many were unaware however that SEELE was behind the scenes, pulling it's strings.
SEELE and the Second Impact
The Second Impact was set about intentionally by SEELE, following the prescribed prophecies detailed in the Dead Sea Scrolls. SEELE's main goal here was to reduce Adam to a more manageable embryonic state, in order to minimize any contact between Adam and its progeny, the Angels. Whose arrival, fifteen years later, was prophesized by those aforementioned Scrolls. SEELE decided and executed this plan, knowing full well of the billions casualties to world population and irreparable changes that would affect the entire planet.
SEELE wishes to be merged inside an Evangelion unit with its own S², after destroying the Spear of Longinus. By placing their own souls within the body of a godlike being, and destroying the only weapon that can harm the body, they would effectively create an immortal God with a human soul which would represent the pinnacle of human existence, and live through all of eternity. --via Eva Fandom Wiki