A search for long-lived, massive particles predicted by many theories beyond the Standard Model is presented. The search targets final states with large missing transverse momentum and at least one high-mass displaced vertex with five or more tracks, and uses 32.8 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of $s$ = 13 TeV $pp$ collision data collected by the ATLAS detector at the LHC. The observed yield is consistent with the expected background. The results are used to extract 95% CL exclusion limits on the production of long-lived gluinos with masses up to 2.37 TeV and lifetimes of $O(10-2)$-$O(10)$ ns in a simplified model inspired by Split Supersymmetry.