A novel scheme is proposed to generate uniform relativistic electron layers for coherent Thomson backscattering. A few-cycle laser pulse is used to produce the electron layer from an ultrathin solid foil. The key element of the new scheme is an additional foil that reflects the drive-laser pulse, but lets the electrons pass almost unperturbed. Making use of two-dimensional particle-in-cell simulations and well-known basic theory, it is shown that the electrons, after interacting with both the drive and reflected laser pulses, form a very uniform flyer freely cruising with a high relativistic γ factor exactly in the drive-laser direction (no transverse momentum). It backscatters the probe light with a full Doppler shift factor of 4γ(2). The reflectivity and its decay due to layer expansion are discussed.