  We prove the existence of a family of compact subdomains $\Omega$ of the flat
cylinder
  $\mathbb{R}^N\times \mathbb{R}/2\pi\mathbb{Z}$ for which the Neumann
eigenvalue problem for the Laplacian on $\Omega$ admits eigenfunctions with
constant Dirichlet values on $\partial \Omega$. These domains $\Omega$ have the
property that their boundaries $\partial \Omega$ have nonconstant principal
curvatures. In the context of ambient Riemannian manifolds, our construction
provides the first examples of such domains whose boundaries are neither
homogeneous nor isoparametric hypersurfaces. The functional analytic approach
we develop in this paper overcomes an inherent loss of regularity of the
problem in standard function spaces. With the help of this approach, we also
construct a related family of subdomains of the $2$-sphere $S^2$. By this we
disprove a conjecture in \cite{Souam}.
