This is not really an answer because these PhD's were never actually written, but anyway: in his book A mathematicians miscellany (in the chapter on math with minimum raw material) Littlewood gave 2 examples that could have been 2-line PhDs:

(1) Cayley's projective definition of length

(2)Theorem: An integral function never 0 or 1 is a constant. Proof: $\exp(i\Omega(f(z)))$ is a bounded integral function. ($\Omega$ is inverse to the elliptic modular function.)