Converting a Gaussian to a flat-top beam is useful for many applications including laser-launched thin-foil flyer plates. A flat-top beam is needed to maintain a constant launch velocity across the flyer; otherwise, the flyer can disintegrate in flight. Here we discuss and demonstrate the use of a variable reflectivity mirror (VRM) with a Gaussian reflectivity profile with an additional hard aperture and compare it to a refractive beam shaper. An ideal VRM would generate a flat-top beam with 37% efficiency. Readily available high-power Gaussian or super-Gaussian mirrors create an approximate flat-top profile, but there is a trade-off between flatness and efficiency. We show that a super-Gaussian mirror can, in principle, convert an input Gaussian beam with 30% efficiency to a flat-top beam with 3% (maximum-to-minimum) variation. With a Gaussian mirror and a high-energy pulsed Nd:YAG laser having relatively poor beam quality, we generate flat-top beams with 25% conversion efficiency having 6% variation (standard deviation sigma=4.2%). The beams are used to launch 400 microm diameter, 25 microm thick Al flyer plates, whose flight was monitored by a high-speed displacement interferometer. The plates flew across a 300 microm gap at 1.3 km/s. The distribution of arrival times at the witness plate was 5 ns, as determined by the rise time of the impact emission. Compared to a total flight time of 260 ns, the velocity spread of different parts of the flyer plate was 2%.