A "double-dome" is an intellectual. I encountered the term in the first episode of Rocky & Bullwinkle, from 1959 or 1960, but it dates back to 1938.

I can see where "egghead" comes from, but I'm not sure about "double-dome." I was envisioning bald aliens with two-lobed heads (like Ferengi or something), but now I'm thinking maybe it just refers to people with high foreheads--double the height/amount of "dome" that normal people would have.