The intense radiation flux of Type I X-ray bursts is expected to interact with the accretion flow around neutron stars. High frequency quasiperiodic oscillations (kHz QPOs), observed at frequencies matching orbital frequencies at tens of gravitational radii, offer a unique probe of the innermost disk regions. In this paper, we follow the lower kHz QPOs, in response to Type I X-ray bursts, in two prototypical QPO sources, namely 4U 1636-536 and 4U 1608-522, as observed by the Proportional Counter Array of the Rossi X-ray Timing Explorer. We have selected a sample of 15 bursts for which the kHz QPO frequency can be tracked on timescales commensurable with the burst durations (tens of seconds). We find evidence that the QPOs are affected for over ∼200 s during one exceptionally long burst and ∼100 s during two others (although at a less significant level), while the burst emission has already decayed to a level that would enable the pre-burst QPO to be detected. On the other hand, for most of our burst-kHz QPO sample, we show that the QPO is detected as soon as the statistics allow and in the best cases, we are able to set an upper limit of ∼20 s on the recovery time of the QPO. This diversity of behavior cannot be related to differences in burst peak luminosity. We discuss these results in the framework of recent findings that accretion onto the neutron star may be enhanced during Type I X-ray bursts. The subsequent disk depletion could explain the disappearance of the QPO for ∼100 s, as possibly observed in two events. However, alternative scenarios would have to be invoked for explaining the short recovery timescales inferred from most bursts. Heating of the innermost disk regions would be a possibility, although we cannot exclude that the burst does not affect the QPO emission at all. Clearly the combination of fast timing and spectral information of Type I X-ray bursts holds great potential in the study of the dynamics of the inner accretion flow around neutron stars. However, as we show, breakthrough observations will require a timing instrument providing at least ten times the effective area of the RXTE/PCA.