I believe that sieve methods (in particular methods similar to Chen's result on "almost" Goldbach and twin primes) should give infinitely primes $p$ such $(p-1)/2$ is a product of two primes. That's more than what you asked.
Wikipedia quotes a result of Bombieri-Friedlander-Iwaniec stating that Linnik's constant is $2$ for almost all moduli. If the same is true for infinitely many prime moduli $q$, then you are in business. A prime $p \equiv 1 \mod q, p \ll q^2$ is what you want.
Maybe an analytic number theorist will come along and give precise references.