Isolated multi-MeV $\gamma$-rays with attosecond duration, high collimation and beam angular momentum (BAM) may find many interesting applications in nuclear physics, astrophysics, etc. Here, we propose a scheme to generate such $\gamma$-rays via nonlinear Thomson scattering of a rotating relativistic electron sheet driven by a few-cycle twisted laser pulse interacting with a micro-droplet target. Our model clarifies the laser intensity threshold and carrier-envelope phase effect on the generation of the isolated electron sheet. Three-dimensional numerical simulations demonstrate the $\gamma$-ray emission with 320 attoseconds duration and peak brilliance of $9.3\times 10^{24}$ photons s${}^{-1}$ mrad${}^{-2}$ mm${}^{-2}$ per 0.1$\%$ bandwidth at 4.3 MeV. The $\gamma$-ray beam carries a large BAM of $2.8 \times 10^{16}\mathrm{\hslash}$, which arises from the efficient BAM transfer from the rotating electron sheet, subsequently leading to a unique angular distribution. This work should promote the experimental investigation of nonlinear Thomson scattering of rotating electron sheets in large laser facilities.