We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff--Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $\omega^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities~$\omega^0_\epsilon\in C^\infty_c(\mathbb{R}^2)$ converging to $\omega^0_{\mathrm{sing}}$ distributionally as $\epsilon\to 0^+$, and show that the corresponding solutions $\omega_\epsilon(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff--Rott system with initial datum $\omega^0_{\mathrm{sing}}$. The regularization relies on a layer construction designed to exploit the key observation that the Kelvin--Helmholtz instability has a strongly anisotropic effect: while vorticities must be analytic in the "tangential" direction, the way layers can be arranged in the "normal" direction is essentially arbitrary.