Maybe I ought to get out more, but I've just spent a very enjoyable hour browsing the Phillipi Collection website which is described as being:
...currently the world’s largest collection of clerical, ecclesiastical and religious head coverings and is unique in both its scope and size. Whether they are ceremonial or worn as a part of everyday life, you’ll find examples of headwear from every religious persuasion around the globe. In addition, the collection includes 116 religious objects.The site has everything you ever wanted to know about borlas, boat-hats, galeros and, of course, birettas galore (who would have guessed that there were so many different types, including tiny teddy-bear sized ones). I was interested to see that a toque is a kind of academic headgear. I spent some years growing up in Canada where "toque" was a common term for a much less elegant woolly winter hat.
There's a great dry sense of humour in the commentary: I particularly liked the understatement in the description of the camouflage biretta: "Indeed a strange biretta!" and the inclusion of Biretta art: the amazing levitating Biretta installation (pictured above).
For the serious aficionado (or for those mothers being pestered by children wanting miniature vestments) there are patterns for birettas -- both collapsible and and the ordinary sort. The site recommends an Italian supplier for the silk pompoms on top, but for children I reckon that a DIY version made by the children themselves will do the trick.