Thomson scattering (TS) from electron beams produced in laser-plasma accelerators may generate femtosecond pulses of quasi-monochromatic, multi-MeV photons. Scaling laws suggest that reaching the necessary GeV electron energy, with a percent-scale energy spread and five-dimensional brightness over 10 16 A/m 2 , requires acceleration in centimeter-length, tenuous plasmas (n 0 $ 10 17 cm À3 ), with petawatt-class lasers. Ultrahigh per-pulse power mandates single-shot operation, frustrating applications dependent on dosage. To generate high-quality near-GeV beams at a manageable average power (thus affording kHz repetition rate), we propose acceleration in a cavity ofelectrondensity ,drivenwithanincoherent stack of sub-Joule laser pulses through a millimeter-length, dense plasma (n 0 $ 10 19 cm
À3). Blue-shifting one stack component by a considerable fraction of the carrier frequency compensates for the frequency red shift imparted by the wake. This avoids catastrophic self-compression of the optical driver and suppresses expansion of the accelerating cavity, avoiding accumulation of a massive low-energy background. In addition, the energy gain doubles compared to the predictions of scaling laws. Head-on collision of the resulting ultrabright beams with another optical pulse produces, via TS, gigawatt γ-ray pulses having a sub-20% bandwidth, over 10 6 photons in a microsteradian observation cone, and the observation cone, and the mean energy tunable up to 16 MeV.