Generating quasi-monochromatic, femtosecond γ-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 mmsize, dense plasma (n 0 ∼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. Blueshifting 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 component of the stack 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.The production of multi-picosecond TS γ-ray pulses has been earlier demonstrated using e-beams from conventional accelerators [12][13][14][15][16][17][18][19][20][21]. These pulses have a high degree of polarization, and are thus attractive as e-beam diagnostics [12,13]. They are also employed in the generation of polarized positrons from dense targets [15] and to demonstrate nuclear fluorescence [17][18][19]21]. Conventional accelerators, however, are large and expensive, which makes linac-based radiation sources scarce and busy user facilities. Also, the large (cm-scale) size of the radio-frequency powered acceleration cavities makes it difficult to produce and synchronize e-beams (and, hence, TS γ-ray pulses) on a sub-ps time scale relevant to high-energy density physics [22]. Luckily, an alternative technical solution, a miniature laser-plasma accelerator (LPA) [23,24], enables production of even shorter (viz. femtosecond) e-beams [25]. Besides, polychromatic (or 'comb-like') beams from an LPA, with the current modulated on a femtosecond scale, have been observed in experiments [26][27][28][29]. Simulations indicate that such beams readily lend themselves to all-optical manipulation, promising generation of spectrally controlled quasi-monochromatic, femtosecond γ-ray pulses, or trains of pulses with a femtosecond synchronization [9][10][11].LPAs, however, face a number of challenges, one of which is preservation of beam quality, that is, elimination of a high-charge, low-energy tail, which develops when acceleration is...