We report measurements of transition energies for highly ionized uranium. For hydrogenic uranium we have measured 102 130\ifmmode\pm\else\textpm\fi{}92 eV for the 2${\mathit{p}}_{3/2\mathrm{\ensuremath{-}}}$1${\mathit{s}}_{1/2}$ transition. This measurement is sensitive to the 1s Lamb shift, probing the electron self-energy in high-field QED. In addition, we have measured the (1s2p${)}^{1}$${\mathit{P}}_{1}$${,}^{3}$${\mathit{P}}_{2\mathrm{\ensuremath{-}}}$(1${\mathit{s}}^{2}$${)}^{1}$${\mathit{S}}_{0}$ transition in heliumlike uranium as 100 598\ifmmode\pm\else\textpm\fi{}111 eV, probing relativistic electron-electron correlation effects. Both measurements were obtained with a Doppler spectrometer, a proven technique which we have adapted to the field of high-Z precision atomic transition-energy measurements. These measurements demonstrate the potential of this technique to attain \ensuremath{\sim}1 eV uncertainties.