We study the exceptionally short (32-41 ms) precursors of two intermediate-duration thermonuclear X-ray bursts observed with RXTE from the neutron stars in 4U 0614+09 and 2S 0918-549. They exhibit photon fluxes that surpass those at the Eddington limit later in the burst by factors of 2.6 to 3.1. We are able to explain both the short duration and the super-Eddington flux by mildly relativistic outflow velocities of 0.1$c$ to 0.3$c$ subsequent to the thermonuclear shell flashes on the neutron stars. These are the highest velocities ever measured from any thermonuclear flash. The precursor rise times are also exceptionally short: about 1 ms. This is inconsistent with predictions for nuclear flames spreading laterally as deflagrations and suggests detonations instead. This is the first time that a detonation is suggested for such a shallow ignition column depth ($y_{rm ign}$ = 10$^{10}$ g cm$^{-2}$). The detonation would possibly require a faster nuclear reaction chain, such as bypassing the alpha-capture on $^{12}$C with the much faster $^{12}$C(p,$gamma$)$^{13}$N($alpha$,p)$^{16}$O process previously proposed. We confirm the possibility of a detonation, albeit only in the radial direction, through the simulation of the nuclear burning with a large nuclear network and at the appropriate ignition depth, although it remains to be seen whether the Zeldovich criterion is met. A detonation would also provide the fast flame spreading over the surface of the neutron star to allow for the short rise times. (...) As an alternative to the detonation scenario, we speculate on the possibility that the whole neutron star surface burns almost instantly in the auto-ignition regime. This is motivated by the presence of 150 ms precursors with 30 ms rise times in some superexpansion bursts from 4U 1820-30 at low ignition column depths of ~10$^8$ g cm$^{-2}$.
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