Two papers on this topic are: Granville's Refinements of Goldbach’s conjecture, and the generalized Riemann hypothesis and Wirsing's Thin subbases.

In Granville's paper it is shown that if a quantitative form of Goldbach's conjecture is true, then there exists a set of primes, B, such that $B \cap [N] \ll (N \ln(N))^{1/2}$ and every even integer is the sum of two elements of B. In Wirsing's paper it is shown that For any $k \geq 3$ there is a set $B_{k}$ of primes, such that $B_{k} \cap [N] \ll (N \ln(N))^{1/k}$, that is a basis of order k for large $n \equiv k mod 2$. All of these constructions are probabilistic.

If you only care that the sumset is almost all of the even/odd integers, one can remove the $\ln^{1/k}(N)$ term in the above results.

Two papers on this topic are: Granville's Refinements of Goldbach’s conjecture, and the generalized Riemann hypothesis and Wirsing's Thin subbases.

In Granville's paper it is shown that if a quantitative form of Goldbach's conjecture is true, then there exists a set of primes, B, such that $|B \cap [N]| \ll (N \ln(N))^{1/2}$ and every even integer is the sum of two elements of B. In Wirsing's paper it is shown that For any $k \geq 3$ there is a set $B_{k}$ of primes, such that $|B_{k} \cap [N]| \ll (N \ln(N))^{1/k}$, that is a basis of order k for large $n \equiv k mod 2$. All of these constructions are probabilistic.

If you only care that the sumset is almost all of the even/odd integers, one can remove the $\ln^{1/k}(N)$ term in the above results.