"Half a teaspoon" treats the teaspoon as a unit, saying that we should use half of that unit.
"A half teaspoon" treats the half-teaspoon as a unit, saying that we should use one of it.
To answer your question as to which is correct: The traditional unit is the teaspoon, not the half-teaspoon - so the first version is correct. However, measuring spoons come in fractional sizes, and most sets include a half-teaspoon measure, so the second version is correct.
This AE speaker says you can use either one.