Godunov’s Theorem [S.K. Godunov, Mat. Sb. 47 (1959)], stated more than six decades ago, set the framework for understanding the limitations of linear numerical schemes for approximating hyperbolic equations numerically. This theoretical result sets one of the basic requirements for constructing high-order numerical schemes, namely non-linearity. In the present article we are concerned with modifications to essentially-non-oscillatory (ENO) non-linear reconstruction approach, along with fully discrete ADER schemes to derive methods of arbitrary order of accuracy in space and time. Here we extend a recently proposed ENO-ET scheme for one-dimensional problems to two space dimensions with Cartesian meshes. The methods are implemented up to fifth order of accuracy and assessed via three scalar 2D problems, namely the linear advection equation, Burgers equation and a kinematic frontogenesis model used in meteorology. Empirical convergence rates are studied for the classical ENO, classical WENO and the newly proposed ENO-ET. For smooth solutions results from the newly proposed ENO-ET reconstruction scheme are superior to those of conventional ENO in terms of theoretically expected convergence rates and size of errors. Compared to the results obtained with WENO reconstruction, the performance of ENO-ET for second and third orders is superior. For discontinuous solutions, again ENO-ET is superior, in that it captures wave amplitudes more accurately than ENO as accurate as WENO and, unlike ENO, exhibits no spurious oscillations near discontinuities.