You're wondering why you can't find a Miltini recipe online? One trick for finding out historical information is to search archive.org. That quickly turns up a 1956 article [1] with the recipe: a Miltini is a six or seven-to-one Martini [i.e. ratio of spirits to vermouth] with a Miltown substituted for the lemon peel or olive.

The dramatically overwrought article continues: "It is doubtful if any other creation of the mid-20th Century so typifies this frantic, uncertain era in which no man can know what the next hour will bring." and suggests that a hermetically-sealed shaker of Miltinis should be placed in time capsules "so that posterity can know what real cool cats we were and how extraordinary the civilzation we wrought."

Personally, it sounds a bit like an urban legend. As the Federal Reserve Bank of Philadelphia said: "Unlike most drugs, its name was easily pronounceable and, better still, lent itself to punsters and jokesmiths on TV screens in the homes of millions of people. Example: 'Miltown Berle,' or 'The Government is giving out a Miltown with every income tax blank,' or 'Use a Miltown instead of an olive to make a Miltini'".

[1] https://archive.org/details/palm-springs-desert-sun-1956-11-...

[2] https://archive.org/details/sim_business-review-federal-rese...