"A half a cup of [something]"

‘Half a [something]’ is normal in BrEng as well as AmEng, and I see no need to justify it by saying it’s short for ‘half of a’. It is enshrined in sayings such as ‘Half a loaf is better than no bread at all’ and ‘Half a pound of tupenny rice’, where the insertion of ‘of’ would look strange. We describe six as ‘half a dozen’ and not as ‘half of a dozen’ and 30 minutes as ‘half an hour’, and not as ‘half of an hour’. In such formulations, ‘half’ is an adjective, as it also is in this citation from Trollope illustrating the OED’s entry for adjectival ‘half’: ‘Though the lord might be only half a man, Julia walked out from the church every inch a countess.’

"A half a cup" is abbreviated speech for "a half of a cup". (Barrie England argues convincingly against this, but the contraction of of a to a happens enough in casual speech that it is hard to believe that this is not an old instance of this tendency.)

Sometimes of is lazily pronounced a as in "a ton a bricks", and the sequence of a just gets shortened to a.

In England, a "cup of tea" is often abbreviated "a cuppa".

"A half a(n) X" is an Americanism; I suspect it may be somewhat more prevalent in the South, but it is certainly not exclusive to that region. Looking at the Google Ngram for "a half a mile" you see that the frequency peaks at .000013 around 1850-1890 in American English. This is roughly six times the highest frequency of this expression in British English (1940-1965). Here is an 1868 New York magazine for teachers complaining about it.

Some writers and speakers have a careless way of using half as an adjective with an a both before and after it, as "We had gone a half a mile or more" ... . Here, of course, the an or a is superfluous, and should be omitted.

I've left out a number of examples in the quote above, the cited ones of which come from New York, New England, and Virginia. I'm sure the author took great pride in finding the example: "Thousands cannot construct a half a dozen consecutive sentences without violating some rule of grammar."

And if it's not clear from the quote above, just because some Americans use it doesn't mean that it's considered to be correct grammar here.

It seems to me that in this case a half a cup and a half a teaspoon may be driven by the usage of measuring cups and spoons with the specified capacity. These expressions imply using, e.g., a measuring cup holding half a cup, etc.