Generating quasi-monochromatic, femtosecond gamma-ray pulses via Thomson scattering (TS) demands exceptional electron beam (e-beam) quality, such as percent scale energy spread and five-dimensional brightness over 10^16 A/m^2. We show that near-GeV e-beams with these metrics can be accelerated in a cavity of electron density, driven with an incoherent stack of Joule-scale laser pulses through a mm-size, dense plasma (n ~ 10^19 cm^-􀀀3). Changing the time delay, frequency difference, and energy ratio of the stack components controls the e-beam phase space on the femtosecond scale, while the modest energy of the optical driver helps afford kHz-scale repetition rate at manageable average power. Blue-shifting one stack component by a considerable fraction of the carrier frequency makes the stack immune to self-compression. This, in turn, minimizes uncontrolled variation in the cavity shape, suppressing continuous injection of ambient plasma electrons, preserving a single, ultra-bright electron bunch. In addition, weak focusing of the trailing stack component induces periodic injection, generating, in a single shot, a train of bunches with controllable energy spacing and femtosecond synchronization. These designer e-beams, inaccessible to conventional acceleration methods, generate, via TS, gigawatt γ-ray pulses (or multi-color pulse trains) with the mean energy in the range of interest for nuclear photonics (4 – 16 MeV), containing over 10^6 photons within a microsteradian-scale observation cone.