  We compute the optimal constant and characterise the maximisers at all
spatial scales for the Agmon--H\"ormander $L^2$-Fourier adjoint restriction
estimate on the sphere. The maximisers switch back and forth from being
constants to being non-symmetric at the zeros of two Bessel functions. We also
study the stability of this estimate and establish a sharpened version in the
spirit of Bianchi--Egnell. The corresponding stability constant and maximisers
again exhibit a curious intermittent behaviour.
