Recently I came across this expression "eggheads and fatheads". I know the individual meanings of both the words. In which context should I use these two words together?
But it's important to note that the "metaphorical" element in both terms is somewhat obscure. The first alludes to a popular stereotype of intellectuals having high foreheads (hence highbrow). The second draws on the various negative connotations of fat, gross, thick, dense implying stupid.
Using both words in close proximity inevitably draws attention to the fact that there's no obvious opposition/contrast between egg and fat except when conjoined with head. It's my opinion that this constitutes a noticeable "disjunct", so I would advise against doing it unless you want the quirky juxtaposition for stylistic reasons.