A few words on what "BRST quantization" is, from a sum-over-histories perspective. Gauge theory and general relativity both have the property, that mathematically different field configurations can be physically equivalent, because of a symmetry (gauge symmetry, general covariance). So when you do a quantum sum over histories, there is a problem of avoiding redundancy - you don't want the same physical state to be counted multiple times.
One way to avoid this is to single out just one mathematical form of each physical state, by imposing an extra condition (a simple example is
Coulomb gauge); this is a form of "gauge-fixing". BRST works differently; it
adds some fictitious extra fields called ghost fields, which basically compensate for the overcounting, in the amended sum over histories. Once the ghost fields are added, the gauge symmetry is broken, but there's a new "BRST symmetry" which assists calculation.
A
BRST symmetry for Ashtekar variables was worked out almost immediately back in the 1980s, but that's only one step in carrying out the full process of making a quantum theory. I dug through the paper's 50 citations; the most advanced follow-up I found was a
1998 paper from Russia, in which the authors start with the "Hilbert-Palatini" action for general relativity, then change the variables to Ashtekar's new variables, and see what that does to the Hilbert-Palatini formulas. I couldn't find a good follow-up to the 1998 paper, but they highligh as their most interesting conclusion, that the BRST path integral for Ashtekar variables is a contour integral. This is also the case with most formulae in twistor theory, so it's consistent with what people keep saying, that the Ashtekar variables have something in common with the twistor perspective.
