Cartan subalgebras in operator algebras have been an active research area since their introduction in the 1970s (by Vershik and Feldman-Moore for von Neumann algebras, and Renault and Kumjian for C*-algebras). While on the side of von Neumann algebras some deep rigidity results have been established, much less is known about the variety of Cartan subalgebras that a C*-algebra can admit, with only a few exceptions of certain subhomogeneous C*-algebras. In this talk, we discuss another case of C*-algebras whose Cartan subalgebras can be parametrized in full, namely unital essential extensions of continuous functions on the circle by the compact operators (which includes the Toeplitz algebra). As we shall see, in the non-trivial extension case, these C*-algebras admit no C*-diagonals, all of their Cartan subalgebras have the one-point compactification of the natural numbers as their spectrum, and, up to conjugacy, these are parametrized by certain "near symmetries" of the natural numbers. This is joint work with Philipp Sibbel.